Showing 1–44 of 44 results for author: Ramanujan, M S

Algorithms for Euclidean Distance Matrix Completion: Exploiting Proximity to Triviality

Authors: Fedor V. Fomin, Petr A. Golovach, M. S. Ramanujan, Saket Saurabh

Abstract: In the d-Euclidean Distance Matrix Completion (d-EDMC) problem, one aims to determine whether a given partial matrix of pairwise distances can be extended to a full Euclidean distance matrix in d dimensions. This problem is a cornerstone of computational geometry with numerous applications. While classical work on this problem often focuses on exploiting connections to semidefinite programming typ… ▽ More In the d-Euclidean Distance Matrix Completion (d-EDMC) problem, one aims to determine whether a given partial matrix of pairwise distances can be extended to a full Euclidean distance matrix in d dimensions. This problem is a cornerstone of computational geometry with numerous applications. While classical work on this problem often focuses on exploiting connections to semidefinite programming typically leading to approximation algorithms, we focus on exact algorithms and propose a novel distance-from-triviality parameterization framework to obtain tractability results for d-EDMC. We identify key structural patterns in the input that capture entry density, including chordal substructures and coverability of specified entries by fully specified principal submatrices. We obtain: (1) The first fixed-parameter algorithm (FPT algorithm) for d-EDMC parameterized by d and the maximum number of unspecified entries per row/column. This is achieved through a novel compression algorithm that reduces a given instance to a submatrix on O(1) rows (for fixed values of the parameters). (2) The first FPT algorithm for d-EDMC parameterized by d and the minimum number of fully specified principal submatrices whose entries cover all specified entries of the given matrix. This result is also achieved through a compression algorithm. (3) A polynomial-time algorithm for d-EDMC when both d and the minimum fill-in of a natural graph representing the specified entries are fixed constants. This result is achieved by combining tools from distance geometry and algorithms from real algebraic geometry. Our work identifies interesting parallels between EDM completion and graph problems, with our algorithms exploiting techniques from both domains. △ Less

Submitted 19 March, 2026; originally announced March 2026.

Comments: Full version of SoCG '26 paper

Tractable Graph Structures in EFX Orientation

Authors: Václav Blažej, Sushmita Gupta, M. S. Ramanujan, Peter Strulo

Abstract: Since its introduction, envy-freeness up to any good (EFX) has become a fundamental solution concept in fair division of indivisible goods. Its existence remains elusive -- even for four agents with additive utility functions, it is unknown whether an EFX allocation always exists. Unsurprisingly, restricted settings to delineate tractable and intractable cases have been explored. Christadolou, Fia… ▽ More Since its introduction, envy-freeness up to any good (EFX) has become a fundamental solution concept in fair division of indivisible goods. Its existence remains elusive -- even for four agents with additive utility functions, it is unknown whether an EFX allocation always exists. Unsurprisingly, restricted settings to delineate tractable and intractable cases have been explored. Christadolou, Fiat et al.[EC'23] introduced the notion of EFX-orientation, where the agents form the vertices of a graph and the items correspond to edges, and an agent values only the items that are incident to it. The goal is to allocate items to one of the adjacent agents while satisfying the EFX condition. Building on the work of Zeng and Mehta'24, which established a sharp complexity threshold based on the structure of the underlying graph -- polynomial-time solvability for bipartite graphs and NP-hardness for graphs with chromatic number at least three -- we further explore the algorithmic landscape of EFX-orientation using parameterized graph algorithms. Specifically, we show that bipartiteness is a surprisingly stringent condition for tractability: EFX orientation is NP-complete even when the valuations are symmetric, binary and the graph is at most two edge-removals away from being bipartite. Moreover, introducing a single non-binary value makes the problem NP-hard even when the graph is only one edge removal away from being bipartite. We further perform a parameterized analysis to examine structures of the underlying graph that enable tractability. In particular, we show that the problem is solvable in linear time on graphs whose treewidth is bounded by a constant and that the complexity of an instance is closely tied to the sizes of acyclic connected components on its one-valued edges. △ Less

Submitted 18 June, 2025; originally announced June 2025.

When Distances Lie: Euclidean Embeddings in the Presence of Outliers and Distance Violations

Authors: Matthias Bentert, Fedor V. Fomin, Petr A. Golovach, M. S. Ramanujan, Saket Saurabh

Abstract: Distance geometry explores the properties of distance spaces that can be exactly represented as the pairwise Euclidean distances between points in $\mathbb{R}^d$ ($d \geq 1$), or equivalently, distance spaces that can be isometrically embedded in $\mathbb{R}^d$. In this work, we investigate whether a distance space can be isometrically embedded in $\mathbb{R}^d$ after applying a limited number of… ▽ More Distance geometry explores the properties of distance spaces that can be exactly represented as the pairwise Euclidean distances between points in $\mathbb{R}^d$ ($d \geq 1$), or equivalently, distance spaces that can be isometrically embedded in $\mathbb{R}^d$. In this work, we investigate whether a distance space can be isometrically embedded in $\mathbb{R}^d$ after applying a limited number of modifications. Specifically, we focus on two types of modifications: outlier deletion (removing points) and distance modification (adjusting distances between points). The central problem, Euclidean Embedding Editing (EEE), asks whether an input distance space on $n$ points can be transformed, using at most $k$ modifications, into a space that is isometrically embeddable in $\mathbb{R}^d$. We present several fixed-parameter tractable (FPT) and approximation algorithms for this problem. Our first result is an algorithm that solves EEE in time $(dk)^{\mathcal{O}(d+k)} + n^{\mathcal{O}(1)}$. The core subroutine of this algorithm, which is of independent interest, is a polynomial-time method for compressing the input distance space into an equivalent instance of EEE with $\mathcal{O}((dk)^2)$ points. For the special but important case of EEE where only outlier deletions are allowed, we improve the parameter dependence of the FPT algorithm and obtain a running time of $\min\{(d+3)^k, 2^{d+k}\} \cdot n^{\mathcal{O}(1)}$. Additionally, we provide an FPT-approximation algorithm for this problem, which outputs a set of at most $2 \cdot {\rm OPT}$ outliers in time $2^d \cdot n^{\mathcal{O}(1)}$. This 2-approximation algorithm improves upon the previous $(3+\varepsilon)$-approximation algorithm by Sidiropoulos, Wang, and Wang [SODA '17]. Furthermore, we complement our algorithms with hardness results motivating our choice of parameterizations. △ Less

Submitted 24 March, 2025; originally announced March 2025.

Comments: An extended abstract of this paper appears in the proceedings of SoCG 2025

Packing Short Cycles

Authors: Matthias Bentert, Fedor V. Fomin, Petr A. Golovach, Tuukka Korhonen, William Lochet, Fahad Panolan, M. S. Ramanujan, Saket Saurabh, Kirill Simonov

Abstract: Cycle packing is a fundamental problem in optimization, graph theory, and algorithms. Motivated by recent advancements in finding vertex-disjoint paths between a specified set of vertices that either minimize the total length of the paths [Björklund, Husfeldt, ICALP 2014; Mari, Mukherjee, Pilipczuk, and Sankowski, SODA 2024] or request the paths to be shortest [Lochet, SODA 2021], we consider the… ▽ More Cycle packing is a fundamental problem in optimization, graph theory, and algorithms. Motivated by recent advancements in finding vertex-disjoint paths between a specified set of vertices that either minimize the total length of the paths [Björklund, Husfeldt, ICALP 2014; Mari, Mukherjee, Pilipczuk, and Sankowski, SODA 2024] or request the paths to be shortest [Lochet, SODA 2021], we consider the following cycle packing problems: Min-Sum Cycle Packing and Shortest Cycle Packing. In Min-Sum Cycle Packing, we try to find, in a weighted undirected graph, $k$ vertex-disjoint cycles of minimum total weight. Our first main result is an algorithm that, for any fixed $k$, solves the problem in polynomial time. We complement this result by establishing the W[1]-hardness of Min-Sum Cycle Packing parameterized by $k$. The same results hold for the version of the problem where the task is to find $k$ edge-disjoint cycles. Our second main result concerns Shortest Cycle Packing, which is a special case of Min-Sum Cycle Packing that asks to find a packing of $k$ shortest cycles in a graph. We prove this problem to be fixed-parameter tractable (FPT) when parameterized by $k$ on weighted planar graphs. We also obtain a polynomial kernel for the edge-disjoint variant of the problem on planar graphs. Deciding whether Min-Sum Cycle Packing is FPT on planar graphs and whether Shortest Cycle Packing is FPT on general graphs remain challenging open questions. △ Less

Submitted 25 October, 2024; v1 submitted 24 October, 2024; originally announced October 2024.

Routing on Sparse Graphs with Non-metric Costs for the Prize-collecting Travelling Salesperson Problem

Authors: Patrick O'Hara, M. S. Ramanujan, Theodoros Damoulas

Abstract: In many real-world routing problems, decision makers must optimise over sparse graphs such as transportation networks with non-metric costs on the edges that do not obey the triangle inequality. Motivated by finding a sufficiently long running route in a city that minimises the air pollution exposure of the runner, we study the Prize-collecting Travelling Salesperson Problem (Pc-TSP) on sparse gra… ▽ More In many real-world routing problems, decision makers must optimise over sparse graphs such as transportation networks with non-metric costs on the edges that do not obey the triangle inequality. Motivated by finding a sufficiently long running route in a city that minimises the air pollution exposure of the runner, we study the Prize-collecting Travelling Salesperson Problem (Pc-TSP) on sparse graphs with non-metric costs. Given an undirected graph with a cost function on the edges and a prize function on the vertices, the goal of Pc-TSP is to find a tour rooted at the origin that minimises the total cost such that the total prize is at least some quota. First, we introduce heuristics designed for sparse graphs with non-metric cost functions where previous work dealt with either a complete graph or a metric cost function. Next, we develop a branch & cut algorithm that employs a new cut we call the disjoint-paths cost-cover (DPCC) cut. Empirical experiments on two datasets show that our heuristics can produce a feasible solution with less cost than a state-of-the-art heuristic from the literature. On datasets with non-metric cost functions, DPCC is found to solve more instances to optimality than the baseline cutting algorithm we compare against. △ Less

Submitted 14 October, 2024; originally announced October 2024.

Comments: Accepted to ATT'24: Workshop Agents in Traffic and Transportation

On Controlling Knockout Tournaments Without Perfect Information

Authors: Václav Blažej, Sushmita Gupta, M. S. Ramanujan, Peter Strulo

Abstract: Over the last decade, extensive research has been conducted on the algorithmic aspects of designing single-elimination (SE) tournaments. Addressing natural questions of algorithmic tractability, we identify key properties of input instances that enable the tournament designer to efficiently schedule the tournament in a way that maximizes the chances of a preferred player winning. Much of the prior… ▽ More Over the last decade, extensive research has been conducted on the algorithmic aspects of designing single-elimination (SE) tournaments. Addressing natural questions of algorithmic tractability, we identify key properties of input instances that enable the tournament designer to efficiently schedule the tournament in a way that maximizes the chances of a preferred player winning. Much of the prior algorithmic work on this topic focuses on the perfect (complete and deterministic) information scenario, especially in the context of fixed-parameter algorithm design. Our contributions constitute the first fixed-parameter tractability results applicable to more general settings of SE tournament design with potential imperfect information. △ Less

Submitted 29 August, 2024; v1 submitted 27 August, 2024; originally announced August 2024.

Comments: v2: Fix author order

On the Parameterized Complexity of Eulerian Strong Component Arc Deletion

Authors: Václav Blažej, Satyabrata Jana, M. S. Ramanujan, Peter Strulo

Abstract: In this paper, we study the Eulerian Strong Component Arc Deletion problem, where the input is a directed multigraph and the goal is to delete the minimum number of arcs to ensure every strongly connected component of the resulting digraph is Eulerian. This problem is a natural extension of the Directed Feedback Arc Set problem and is also known to be motivated by certain scenarios arising in the… ▽ More In this paper, we study the Eulerian Strong Component Arc Deletion problem, where the input is a directed multigraph and the goal is to delete the minimum number of arcs to ensure every strongly connected component of the resulting digraph is Eulerian. This problem is a natural extension of the Directed Feedback Arc Set problem and is also known to be motivated by certain scenarios arising in the study of housing markets. The complexity of the problem, when parameterized by solution size (i.e., size of the deletion set), has remained unresolved and has been highlighted in several papers. In this work, we answer this question by ruling out (subject to the usual complexity assumptions) a fixed-parameter tractable (FPT) algorithm for this parameter and conduct a broad analysis of the problem with respect to other natural parameterizations. We prove both positive and negative results. Among these, we demonstrate that the problem is also hard (W[1]-hard or even para-NP-hard) when parameterized by either treewidth or maximum degree alone. Complementing our lower bounds, we establish that the problem is in XP when parameterized by treewidth and FPT when parameterized either by both treewidth and maximum degree or by both treewidth and solution size. We show that these algorithms have near-optimal asymptotic dependence on the treewidth assuming the Exponential Time Hypothesis. △ Less

Submitted 3 July, 2025; v1 submitted 25 August, 2024; originally announced August 2024.

Comments: v2: Final version accepted to Algorithmica

Parameterized Algorithms for Coordinated Motion Planning: Minimizing Energy

Authors: Argyrios Deligkas, Eduard Eiben, Robert Ganian, Iyad Kanj, M. S. Ramanujan

Abstract: We study the parameterized complexity of a generalization of the coordinated motion planning problem on graphs, where the goal is to route a specified subset of a given set of $k$ robots to their destinations with the aim of minimizing the total energy (i.e., the total length traveled). We develop novel techniques to push beyond previously-established results that were restricted to solid grids.… ▽ More We study the parameterized complexity of a generalization of the coordinated motion planning problem on graphs, where the goal is to route a specified subset of a given set of $k$ robots to their destinations with the aim of minimizing the total energy (i.e., the total length traveled). We develop novel techniques to push beyond previously-established results that were restricted to solid grids. We design a fixed-parameter additive approximation algorithm for this problem parameterized by $k$ alone. This result, which is of independent interest, allows us to prove the following two results pertaining to well-studied coordinated motion planning problems: (1) A fixed-parameter algorithm, parameterized by $k$, for routing a single robot to its destination while avoiding the other robots, which is related to the famous Rush-Hour Puzzle; and (2) a fixed-parameter algorithm, parameterized by $k$ plus the treewidth of the input graph, for the standard \textsc{Coordinated Motion Planning} (CMP) problem in which we need to route all the $k$ robots to their destinations. The latter of these results implies, among others, the fixed-parameter tractability of CMP parameterized by $k$ on graphs of bounded outerplanarity, which include bounded-height subgrids. We complement the above results with a lower bound which rules out the fixed-parameter tractability for CMP when parameterized by the total energy. This contrasts the recently-obtained tractability of the problem on solid grids under the same parameterization. As our final result, we strengthen the aforementioned fixed-parameter tractability to hold not only on solid grids but all graphs of bounded local treewidth -- a class including, among others, all graphs of bounded genus. △ Less

Submitted 30 January, 2026; v1 submitted 24 April, 2024; originally announced April 2024.

ACM Class: F.2; G.2; I.2

An Exercise in Tournament Design: When Some Matches Must Be Scheduled

Authors: Sushmita Gupta, M. S. Ramanujan, Peter Strulo

Abstract: Single-elimination (SE) tournaments are a popular format used in competitive environments and decision making. Algorithms for SE tournament manipulation have been an active topic of research in recent years. In this paper, we initiate the algorithmic study of a novel variant of SE tournament manipulation that aims to model the fact that certain matchups are highly desired in a sporting context, in… ▽ More Single-elimination (SE) tournaments are a popular format used in competitive environments and decision making. Algorithms for SE tournament manipulation have been an active topic of research in recent years. In this paper, we initiate the algorithmic study of a novel variant of SE tournament manipulation that aims to model the fact that certain matchups are highly desired in a sporting context, incentivizing an organizer to manipulate the bracket to make such matchups take place. We obtain both hardness and tractability results. We show that while the problem of computing a bracket enforcing a given set of matches in an SE tournament is NP-hard, there are natural restrictions that lead to polynomial-time solvability. In particular, we show polynomial-time solvability if there is a linear ordering on the ability of players with only a constant number of exceptions where a player with lower ability beats a player with higher ability. △ Less

Submitted 9 February, 2024; originally announced February 2024.

Comments: Full version of AAAI 2024 paper

Highly Connected Steiner Subgraph -- Parameterized Algorithms and Applications to Hitting Set Problems

Authors: Eduard Eiben, Diptapriyo Majumdar, M. S. Ramanujan

Abstract: Given a simple connected undirected graph G = (V, E), a set X \subseteq V(G), and integers k and p, STEINER SUBGRAPH EXTENSION problem asks if there exists a set S \supseteq X with at most k vertices such that G[S] is p-edge-connected. This is a natural generalization of a well-studied problem STEINER TREE (set p=1 and X as the set of all terminals). In this paper, we initiate the study of STEINER… ▽ More Given a simple connected undirected graph G = (V, E), a set X \subseteq V(G), and integers k and p, STEINER SUBGRAPH EXTENSION problem asks if there exists a set S \supseteq X with at most k vertices such that G[S] is p-edge-connected. This is a natural generalization of a well-studied problem STEINER TREE (set p=1 and X as the set of all terminals). In this paper, we initiate the study of STEINER SUBGRAPH EXTENSION from the perspective of parameterized complexity and give a fixed-parameter algorithm parameterized by k and p on graphs of bounded degeneracy. In case we remove the assumption of the input graph being bounded degenerate, then the STEINER SUBGRAPH EXTENSION problem becomes W[1]-hard. Besides being an independent advance on the parameterized complexity of network design problems, our result has natural applications. In particular, we use our result to obtain singly exponential-time FPT algorithms for several vertex deletion problem studied in the literature, where the goal is to delete a smallest set of vertices such that (i) the resulting graph belongs to a specific hereditary graph class, and (ii) the deleted set of vertices induces a p-edge-connected subgraph of the input graph. △ Less

Submitted 3 October, 2025; v1 submitted 5 November, 2023; originally announced November 2023.

Comments: Accepted for publication in SIAM Journal on Discrete Mathematics (SIDMA). Preliminary version appeared in MFCS-2023. Major revisions have been incorporated after incorporating reviewers' comments

ACM Class: F.2.2; G.2.2

FPT Approximations for Packing and Covering Problems Parameterized by Elimination Distance and Even Less

Authors: Tanmay Inamdar, Lawqueen Kanesh, Madhumita Kundu, M. S. Ramanujan, Saket Saurabh

Abstract: For numerous graph problems in the realm of parameterized algorithms, using the size of a smallest deletion set (called a modulator) into well-understood graph families as parameterization has led to a long and successful line of research. Recently, however, there has been an extensive study of structural parameters that are potentially much smaller than the modulator size. In particular, recent p… ▽ More For numerous graph problems in the realm of parameterized algorithms, using the size of a smallest deletion set (called a modulator) into well-understood graph families as parameterization has led to a long and successful line of research. Recently, however, there has been an extensive study of structural parameters that are potentially much smaller than the modulator size. In particular, recent papers [Jansen et al. STOC 2021; Agrawal et al. SODA 2022] have studied parameterization by the size of the modulator to a graph family $\mathcal{H}$ ($\textbf{mod}_{\mathcal{H}}$), elimination distance to $\mathcal{H}$ ($\textbf{ed}_{\mathcal{H}}$), and $\mathcal{H}$-treewidth ($\textbf{tw}_{\mathcal{H}}$). While these new parameters have been successfully exploited to design fast exact algorithms their utility (especially that of latter two) in the context of approximation algorithms is mostly unexplored. The conceptual contribution of this paper is to present novel algorithmic meta-theorems that expand the impact of these structural parameters to the area of FPT Approximation, mirroring their utility in the design of exact FPT algorithms. Precisely, we show that if a covering or packing problem is definable in Monadic Second Order Logic and has a property called Finite Integer Index, then the existence of an FPT Approximation Scheme (FPT-AS, i.e., ($1\pm ε$)-approximation) parameterized these three parameters is in fact equivalent. As concrete exemplifications of our meta-theorems, we obtain FPT-ASes for well-studied graph problems such as Vertex Cover, Feedback Vertex Set, Cycle Packing and Dominating Set, parameterized by these three parameters. △ Less

Submitted 5 October, 2023; originally announced October 2023.

Comments: Full version of FSTTCS 2023 paper. Abstract shortened to meet the character limit

Meta-theorems for Parameterized Streaming Algorithms

Authors: Daniel Lokshtanov, Pranabendu Misra, Fahad Panolan, M. S. Ramanujan, Saket Saurabh, Meirav Zehavi

Abstract: The streaming model was introduced to parameterized complexity independently by Fafianie and Kratsch [MFCS14] and by Chitnis, Cormode, Hajiaghayi and Monemizadeh [SODA15]. Subsequently, it was broadened by Chitnis, Cormode, Esfandiari, Hajiaghayi and Monemizadeh [SPAA15] and by Chitnis, Cormode, Esfandiari, Hajiaghayi, McGregor, Monemizadeh and Vorotnikova [SODA16]. Despite its strong motivation,… ▽ More The streaming model was introduced to parameterized complexity independently by Fafianie and Kratsch [MFCS14] and by Chitnis, Cormode, Hajiaghayi and Monemizadeh [SODA15]. Subsequently, it was broadened by Chitnis, Cormode, Esfandiari, Hajiaghayi and Monemizadeh [SPAA15] and by Chitnis, Cormode, Esfandiari, Hajiaghayi, McGregor, Monemizadeh and Vorotnikova [SODA16]. Despite its strong motivation, the applicability of the streaming model to central problems in parameterized complexity has remained, for almost a decade, quite limited. Indeed, due to simple $Ω(n)$-space lower bounds for many of these problems, the $k^{O(1)}\cdot {\rm polylog}(n)$-space requirement in the model is too strict. Thus, we explore {\em semi-streaming} algorithms for parameterized graph problems, and present the first systematic study of this topic. Crucially, we aim to construct succinct representations of the input on which optimal post-processing time complexity can be achieved. - We devise meta-theorems specifically designed for parameterized streaming and demonstrate their applicability by obtaining the first $k^{O(1)}\cdot n\cdot {\rm polylog}(n)$-space streaming algorithms for well-studied problems such as Feedback Vertex Set on Tournaments, Cluster Vertex Deletion, Proper Interval Vertex Deletion and Block Vertex Deletion. In the process, we demonstrate a fundamental connection between semi-streaming algorithms for recognizing graphs in a graph class H and semi-streaming algorithms for the problem of vertex deletion into H. - We present an algorithmic machinery for obtaining streaming algorithms for cut problems and exemplify this by giving the first $k^{O(1)}\cdot n\cdot {\rm polylog}(n)$-space streaming algorithms for Graph Bipartitization, Multiway Cut and Subset Feedback Vertex Set. △ Less

Submitted 3 August, 2023; originally announced August 2023.

An Improved Time-Efficient Approximate Kernelization for Connected Treedepth Deletion Set

Authors: Eduard Eiben, Diptapriyo Majumdar, M. S. Ramanujan

Abstract: We study the CONNECTED η-TREEDEPTH DELETION problem where the input instance is an undireted graph G = (V, E) and an integer k. The objective is to decide if G has a set S \subseteq V(G) of at most k vertices such that G - S has treedepth at most ηand G[S] is connected. As this problem naturally generalizes the well-known CONNECTED VERTEX COVER, when parameterized by solution size k, the CONNECTED… ▽ More We study the CONNECTED η-TREEDEPTH DELETION problem where the input instance is an undireted graph G = (V, E) and an integer k. The objective is to decide if G has a set S \subseteq V(G) of at most k vertices such that G - S has treedepth at most ηand G[S] is connected. As this problem naturally generalizes the well-known CONNECTED VERTEX COVER, when parameterized by solution size k, the CONNECTED η-TREEDEPTH DELETION does not admit polynomial kernel unless NP \subseteq coNP/poly. This motivates us to design an approximate kernel of polynomial size for this problem. In this paper, we show that for every 0 < ε<= 1, CONNECTED η-TREEDEPTH DELETION SET admits a (1+ε)-approximate kernel with O(k^{2^{η+ 1/ε}}) vertices, i.e. a polynomial-sized approximate kernelization scheme (PSAKS). △ Less

Submitted 1 December, 2022; originally announced December 2022.

Comments: 19 pages, 1 figures. Preliminary version appeared in Proceedings of WG-2022

On the Complexity of Majority Illusion in Social Networks

Authors: Umberto Grandi, Grzegorz Lisowski, M. S. Ramanujan, Paolo Turrini

Abstract: Majority illusion occurs in a social network when the majority of the network nodes belong to a certain type but each node's neighbours mostly belong to a different type, therefore creating the wrong perception, i.e., the illusion, that the majority type is different from the actual one. From a system engineering point of view, we want to devise algorithms to detect and, crucially, correct this un… ▽ More Majority illusion occurs in a social network when the majority of the network nodes belong to a certain type but each node's neighbours mostly belong to a different type, therefore creating the wrong perception, i.e., the illusion, that the majority type is different from the actual one. From a system engineering point of view, we want to devise algorithms to detect and, crucially, correct this undesirable phenomenon. In this paper we initiate the computational study of majority illusion in social networks, providing complexity results for its occurrence and avoidance. Namely, we show that identifying whether a network can be labelled such that majority illusion is present, as well as the problem of removing an illusion by adding or deleting edges of the network, are NP-complete problems. △ Less

Submitted 4 May, 2022; originally announced May 2022.

Deleting, Eliminating and Decomposing to Hereditary Classes Are All FPT-Equivalent

Authors: Akanksha Agrawal, Lawqueen Kanesh, Daniel Lokshtanov, Fahad Panolan, M. S. Ramanujan, Saket Saurabh, Meirav Zehavi

Abstract: For a graph class ${\cal H}$, the graph parameters elimination distance to ${\cal H}$ (denoted by ${\bf ed}_{\cal H}$) [Bulian and Dawar, Algorithmica, 2016], and ${\cal H}$-treewidth (denoted by ${\bf tw}_{\cal H}$) [Eiben et al. JCSS, 2021] aim to minimize the treedepth and treewidth, respectively, of the "torso" of the graph induced on a modulator to the graph class ${\cal H}$. Here, the torso… ▽ More For a graph class ${\cal H}$, the graph parameters elimination distance to ${\cal H}$ (denoted by ${\bf ed}_{\cal H}$) [Bulian and Dawar, Algorithmica, 2016], and ${\cal H}$-treewidth (denoted by ${\bf tw}_{\cal H}$) [Eiben et al. JCSS, 2021] aim to minimize the treedepth and treewidth, respectively, of the "torso" of the graph induced on a modulator to the graph class ${\cal H}$. Here, the torso of a vertex set $S$ in a graph $G$ is the graph with vertex set $S$ and an edge between two vertices $u, v \in S$ if there is a path between $u$ and $v$ in $G$ whose internal vertices all lie outside $S$. In this paper, we show that from the perspective of (non-uniform) fixed-parameter tractability (FPT), the three parameters described above give equally powerful parameterizations for every hereditary graph class ${\cal H}$ that satisfies mild additional conditions. In fact, we show that for every hereditary graph class ${\cal H}$ satisfying mild additional conditions, with the exception of ${\bf tw}_{\cal H}$ parameterized by ${\bf ed}_{\cal H}$, for every pair of these parameters, computing one parameterized by itself or any of the others is FPT-equivalent to the standard vertex-deletion (to ${\cal H}$) problem. As an example, we prove that an FPT algorithm for the vertex-deletion problem implies a non-uniform FPT algorithm for computing ${\bf ed}_{\cal H}$ and ${\bf tw}_{\cal H}$. The conclusions of non-uniform FPT algorithms being somewhat unsatisfactory, we essentially prove that if ${\cal H}$ is hereditary, union-closed, CMSO-definable, and (a) the canonical equivalence relation (or any refinement thereof) for membership in the class can be efficiently computed, or (b) the class admits a "strong irrelevant vertex rule", then there exists a uniform FPT algorithm for ${\bf ed}_{\cal H}$. △ Less

Submitted 6 January, 2022; v1 submitted 20 April, 2021; originally announced April 2021.

Comments: 70 pages, To appear in SODA 2022. arXiv admin note: text overlap with arXiv:2103.09715, arXiv:2105.04660 by other authors

On the Parameterized Complexity of Deletion to $\mathcal{H}$-free Strong Components

Authors: Rian Neogi, M. S. Ramanujan, Saket Saurabh, Roohani Sharma

Abstract: {\sc Directed Feedback Vertex Set (DFVS)} is a fundamental computational problem that has received extensive attention in parameterized complexity. In this paper, we initiate the study of a wide generalization, the {\sc ${\cal H}$-free SCC Deletion} problem. Here, one is given a digraph $D$, an integer $k$ and the objective is to decide whether there is a vertex set of size at most $k$ whose delet… ▽ More {\sc Directed Feedback Vertex Set (DFVS)} is a fundamental computational problem that has received extensive attention in parameterized complexity. In this paper, we initiate the study of a wide generalization, the {\sc ${\cal H}$-free SCC Deletion} problem. Here, one is given a digraph $D$, an integer $k$ and the objective is to decide whether there is a vertex set of size at most $k$ whose deletion leaves a digraph where every strong component excludes graphs in the fixed finite family ${\cal H}$ as (not necessarily induced) subgraphs. When ${\cal H}$ comprises only the digraph with a single arc, then this problem is precisely DFVS. Our main result is a proof that this problem is fixed-parameter tractable parameterized by the size of the deletion set if ${\cal H}$ only contains rooted graphs or if ${\cal H}$ contains at least one directed path. Along with generalizing the fixed-parameter tractability result for DFVS, our result also generalizes the recent results of Göke et al. [CIAC 2019] for the {\sc 1-Out-Regular Vertex Deletion} and {\sc Bounded Size Strong Component Vertex Deletion} problems. Moreover, we design algorithms for the two above mentioned problems, whose running times are better and match with the best bounds for {\sc DFVS}, without using the heavy machinery of shadow removal as is done by Göke et al. [CIAC 2019]. △ Less

Submitted 24 August, 2020; v1 submitted 4 May, 2020; originally announced May 2020.

On the Constrained Least-cost Tour Problem

Authors: Patrick O'Hara, M. S. Ramanujan, Theodoros Damoulas

Abstract: We introduce the Constrained Least-cost Tour (CLT) problem: given an undirected graph with weight and cost functions on the edges, minimise the total cost of a tour rooted at a start vertex such that the total weight lies within a given range. CLT is related to the family of Travelling Salesman Problems with Profits, but differs by defining the weight function on edges instead of vertices, and by… ▽ More We introduce the Constrained Least-cost Tour (CLT) problem: given an undirected graph with weight and cost functions on the edges, minimise the total cost of a tour rooted at a start vertex such that the total weight lies within a given range. CLT is related to the family of Travelling Salesman Problems with Profits, but differs by defining the weight function on edges instead of vertices, and by requiring the total weight to be within a range instead of being at least some quota. We prove CLT is $\mathcal{NP}$-hard, even in the simple case when the input graph is a path. We derive an informative lower bound by relaxing the integrality of edges and propose a heuristic motivated by this relaxation. For the case that requires the tour to be a simple cycle, we develop two heuristics which exploit Suurballe's algorithm to find low-cost, weight-feasible cycles. We demonstrate our algorithms by addressing a real-world problem that affects urban populations: finding routes that minimise air pollution exposure for walking, running and cycling in the city of London. △ Less

Submitted 18 June, 2019; originally announced June 2019.

On the Approximate Compressibility of Connected Vertex Cover

Authors: Diptapriyo Majumdar, M. S. Ramanujan, Saket Saurabh

Abstract: The Connected Vertex Cover problem, where the goal is to compute a minimum set of vertices in a given graph which forms a vertex cover and induces a connected subgraph, is a fundamental combinatorial problem and has received extensive attention in various subdomains of algorithmics. In the area of kernelization, it is known that this problem is unlikely to have efficient preprocessing algorithms,… ▽ More The Connected Vertex Cover problem, where the goal is to compute a minimum set of vertices in a given graph which forms a vertex cover and induces a connected subgraph, is a fundamental combinatorial problem and has received extensive attention in various subdomains of algorithmics. In the area of kernelization, it is known that this problem is unlikely to have efficient preprocessing algorithms, also known as polynomial kernelizations. However, it has been shown in a recent work of Lokshtanov et al. [STOC 2017] that if one considered an appropriate notion of approximate kernelization, then this problem parameterized by the solution size does admit an approximate polynomial kernelization. In fact, Lokhtanov et al. were able to obtain a polynomial size approximate kernelization scheme (PSAKS) for Connected Vertex Cover parameterized by the solution size. A PSAKS is essentially a preprocessing algorithm whose error can be made arbitrarily close to 0. In this paper we revisit this problem, and consider parameters that are strictly smaller than the size of the solution and obtain the first polynomial size approximate kernelization schemes for the Connected Vertex Cover problem when parameterized by the deletion distance of the input graph to the class of cographs, the class of bounded treewidth graphs, and the class of all chordal graphs. △ Less

Submitted 28 April, 2020; v1 submitted 8 May, 2019; originally announced May 2019.

Comments: 1 figure; Revisions from the previous version incorporated based on the comments from some anonymous reviewers

ACM Class: F.2; G.2.2

Journal ref: Algorithmica, 2020

On the Complexity Landscape of Connected f -Factor Problems

Authors: R. Ganian, N. S. Narayanaswamy, S. Ordyniak, C. S. Rahul, M. S. Ramanujan

Abstract: Let G be an undirected simple graph having n vertices and let f be a function defined to be f:V(G) -> {0,..., n-1}. An f-factor of G is a spanning subgraph H such that degree of a vertex v in H is f(v) for every vertex v in V(G). The subgraph H is called a connected f-factor if, in addition, H is connected. A classical result of Tutte(1954) is the polynomial time algorithm to check whether a given… ▽ More Let G be an undirected simple graph having n vertices and let f be a function defined to be f:V(G) -> {0,..., n-1}. An f-factor of G is a spanning subgraph H such that degree of a vertex v in H is f(v) for every vertex v in V(G). The subgraph H is called a connected f-factor if, in addition, H is connected. A classical result of Tutte(1954) is the polynomial time algorithm to check whether a given graph has a specified f-factor. However, checking for the presence of a connected f-factor is easily seen to generalize HAMILTONIAN CYCLE and hence is NP-complete. In fact, the CONNECTED f-FACTOR problem remains NP-complete even when we restrict f(v) to be at least n^e for each vertex v and 0<e<1; on the other side of the spectrum of nontrivial lower bounds on f, the problem is known to be polynomial time solvable when f(v) is at least n/3 for every vertex v. In this paper, we extend this line of work and obtain new complexity results based on restrictions on the function f. In particular, we show that when f(v) is restricted to be at least n/(log n)^c , the problem can be solved in quasi-polynomial time in general and in randomized polynomial time if c<=1. Furthermore, we show that when c>1, the problem is NP-intermediate. △ Less

Submitted 5 December, 2018; originally announced December 2018.

Comments: Under review in Algorithmica

A New Perspective on FO Model Checking of Dense Graph Classes

Authors: Jakub Gajarský, Petr Hliněný, Daniel Lokshtanov, Jan Obdržálek, M. S. Ramanujan

Abstract: We study the first-order (FO) model checking problem of dense graphs, namely those which have FO interpretations in (or are FO transductions of) some sparse graph classes. We give a structural characterization of the graph classes which are FO interpretable in graphs of bounded degree. This characterization allows us to efficiently compute such an FO interpretation for an input graph. As a consequ… ▽ More We study the first-order (FO) model checking problem of dense graphs, namely those which have FO interpretations in (or are FO transductions of) some sparse graph classes. We give a structural characterization of the graph classes which are FO interpretable in graphs of bounded degree. This characterization allows us to efficiently compute such an FO interpretation for an input graph. As a consequence, we obtain an FPT algorithm for successor-invariant FO model checking of any graph class which is FO interpretable in (or an FO transduction of) a graph class of bounded degree. The approach we use to obtain these results may also be of independent interest. △ Less

Submitted 4 May, 2018; originally announced May 2018.

Alternative parameterizations of Metric Dimension

Authors: Gregory Gutin, M. S. Ramanujan, Felix Reidl, Magnus Wahlström

Abstract: A set of vertices $W$ in a graph $G$ is called resolving if for any two distinct $x,y\in V(G)$, there is $v\in W$ such that ${\rm dist}_G(v,x)\neq{\rm dist}_G(v,y)$, where ${\rm dist}_G(u,v)$ denotes the length of a shortest path between $u$ and $v$ in the graph $G$. The metric dimension ${\rm md}(G)$ of $G$ is the minimum cardinality of a resolving set. The Metric Dimension problem, i.e. deciding… ▽ More A set of vertices $W$ in a graph $G$ is called resolving if for any two distinct $x,y\in V(G)$, there is $v\in W$ such that ${\rm dist}_G(v,x)\neq{\rm dist}_G(v,y)$, where ${\rm dist}_G(u,v)$ denotes the length of a shortest path between $u$ and $v$ in the graph $G$. The metric dimension ${\rm md}(G)$ of $G$ is the minimum cardinality of a resolving set. The Metric Dimension problem, i.e. deciding whether ${\rm md}(G)\le k$, is NP-complete even for interval graphs (Foucaud et al., 2017). We study Metric Dimension (for arbitrary graphs) from the lens of parameterized complexity. The problem parameterized by $k$ was proved to be $W[2]$-hard by Hartung and Nichterlein (2013) and we study the dual parameterization, i.e., the problem of whether ${\rm md}(G)\le n- k,$ where $n$ is the order of $G$. We prove that the dual parameterization admits (a) a kernel with at most $3k^4$ vertices and (b) an algorithm of runtime $O^*(4^{k+o(k)}).$ Hartung and Nichterlein (2013) also observed that Metric Dimension is fixed-parameter tractable when parameterized by the vertex cover number $vc(G)$ of the input graph. We complement this observation by showing that it does not admit a polynomial kernel even when parameterized by $vc(G) + k$. Our reduction also gives evidence for non-existence of polynomial Turing kernels. △ Less

Submitted 27 April, 2018; originally announced April 2018.

Reducing CMSO Model Checking to Highly Connected Graphs

Authors: Daniel Lokshtanov, M. S. Ramanujan, Saket Saurabh, Meirav Zehavi

Abstract: Given a Counting Monadic Second Order (CMSO) sentence $ψ$, the CMSO$[ψ]$ problem is defined as follows. The input to CMSO$[ψ]$ is a graph $G$, and the objective is to determine whether $G\models ψ$. Our main theorem states that for every CMSO sentence $ψ$, if CMSO$[ψ]$ is solvable in polynomial time on "globally highly connected graphs", then CMSO$[ψ]$ is solvable in polynomial time (on general gr… ▽ More Given a Counting Monadic Second Order (CMSO) sentence $ψ$, the CMSO$[ψ]$ problem is defined as follows. The input to CMSO$[ψ]$ is a graph $G$, and the objective is to determine whether $G\models ψ$. Our main theorem states that for every CMSO sentence $ψ$, if CMSO$[ψ]$ is solvable in polynomial time on "globally highly connected graphs", then CMSO$[ψ]$ is solvable in polynomial time (on general graphs). We demonstrate the utility of our theorem in the design of parameterized algorithms. Specifically we show that technical problem-specific ingredients of a powerful method for designing parameterized algorithms, recursive understanding, can be replaced by a black-box invocation of our main theorem. We also show that our theorem can be easily deployed to show fixed parameterized tractability of a wide range of problems, where the input is a graph $G$ and the task is to find a connected induced subgraph of $G$ such that "few" vertices in this subgraph have neighbors outside the subgraph, and additionally the subgraph has a CMSO-definable property. △ Less

Submitted 5 February, 2018; originally announced February 2018.

On Structural Parameterizations of the Edge Disjoint Paths Problem

Authors: Robert Ganian, Sebastian Ordyniak, M. S. Ramanujan

Abstract: In this paper we revisit the classical Edge Disjoint Paths (EDP) problem, where one is given an undirected graph G and a set of terminal pairs P and asks whether G contains a set of pairwise edge-disjoint paths connecting every terminal pair in P. Our focus lies on structural parameterizations for the problem that allow for efficient (polynomial-time or fpt) algorithms. As our first result, we ans… ▽ More In this paper we revisit the classical Edge Disjoint Paths (EDP) problem, where one is given an undirected graph G and a set of terminal pairs P and asks whether G contains a set of pairwise edge-disjoint paths connecting every terminal pair in P. Our focus lies on structural parameterizations for the problem that allow for efficient (polynomial-time or fpt) algorithms. As our first result, we answer an open question stated in Fleszar, Mnich, and Spoerhase (2016), by showing that the problem can be solved in polynomial time if the input graph has a feedback vertex set of size one. We also show that EDP parameterized by the treewidth and the maximum degree of the input graph is fixed-parameter tractable. Having developed two novel algorithms for EDP using structural restrictions on the input graph, we then turn our attention towards the augmented graph, i.e., the graph obtained from the input graph after adding one edge between every terminal pair. In constrast to the input graph, where EDP is known to remain NP-hard even for treewidth two, a result by Zhou et al. (2000) shows that EDP can be solved in non-uniform polynomial time if the augmented graph has constant treewidth; we note that the possible improvement of this result to an fpt-algorithm has remained open since then. We show that this is highly unlikely by establishing the W[1]-hardness of the problem parameterized by the treewidth (and even feedback vertex set) of the augmented graph. Finally, we develop an fpt-algorithm for EDP by exploiting a novel structural parameter of the augmented graph. △ Less

Submitted 6 November, 2017; originally announced November 2017.

Comments: Accepted for ISAAC 2017

Saving Critical Nodes with Firefighters is FPT

Authors: Jayesh Choudhari, Anirban Dasgupta, Neeldhara Misra, M. S. Ramanujan

Abstract: We consider the problem of firefighting to save a critical subset of nodes. The firefighting game is a turn-based game played on a graph, where the fire spreads to vertices in a breadth-first manner from a source, and firefighters can be placed on yet unburnt vertices on alternate rounds to block the fire. In this work, we consider the problem of saving a critical subset of nodes from catching fir… ▽ More We consider the problem of firefighting to save a critical subset of nodes. The firefighting game is a turn-based game played on a graph, where the fire spreads to vertices in a breadth-first manner from a source, and firefighters can be placed on yet unburnt vertices on alternate rounds to block the fire. In this work, we consider the problem of saving a critical subset of nodes from catching fire, given a total budget on the number of firefighters. We show that the problem is para-NP-hard when parameterized by the size of the critical set. We also show that it is fixed-parameter tractable on general graphs when parameterized by the number of firefighters. We also demonstrate improved running times on trees and establish that the problem is unlikely to admit a polynomial kernelization (even when restricted to trees). Our work is the first to exploit the connection between the firefighting problem and the notions of important separators and tight separator sequences. Finally, we consider the spreading model of the firefighting game, a closely related problem, and show that the problem of saving a critical set parameterized by the number of firefighters is W[2]-hard, which contrasts our FPT result for the non-spreading model. △ Less

Submitted 30 May, 2017; originally announced May 2017.

Comments: 21 pages, Accepted at ICALP-2017

Path-contractions, edge deletions and connectivity preservation

Authors: Gregory Gutin, M. S. Ramanujan, Felix Reidl, Magnus Wahlström

Abstract: We study several problems related to graph modification problems under connectivity constraints from the perspective of parameterized complexity: {\sc (Weighted) Biconnectivity Deletion}, where we are tasked with deleting~$k$ edges while preserving biconnectivity in an undirected graph, {\sc Vertex-deletion Preserving Strong Connectivity}, where we want to maintain strong connectivity of a digraph… ▽ More We study several problems related to graph modification problems under connectivity constraints from the perspective of parameterized complexity: {\sc (Weighted) Biconnectivity Deletion}, where we are tasked with deleting~$k$ edges while preserving biconnectivity in an undirected graph, {\sc Vertex-deletion Preserving Strong Connectivity}, where we want to maintain strong connectivity of a digraph while deleting exactly~$k$ vertices, and {\sc Path-contraction Preserving Strong Connectivity}, in which the operation of path contraction on arcs is used instead. The parameterized tractability of this last problem was posed by Bang-Jensen and Yeo [DAM 2008] as an open question and we answer it here in the negative: both variants of preserving strong connectivity are $\sf W[1]$-hard. Preserving biconnectivity, on the other hand, turns out to be fixed parameter tractable and we provide a $2^{O(k\log k)} n^{O(1)}$-algorithm that solves {\sc Weighted Biconnectivity Deletion}. Further, we show that the unweighted case even admits a randomized polynomial kernel. All our results provide further interesting data points for the systematic study of connectivity-preservation constraints in the parameterized setting. △ Less

Submitted 21 April, 2017; originally announced April 2017.

Parameterized Complexity and Approximability of Directed Odd Cycle Transversal

Authors: Daniel Lokshtanov, M. S. Ramanujan, Saket Saurabh, Meirav Zehavi

Abstract: A directed odd cycle transversal of a directed graph (digraph) $D$ is a vertex set $S$ that intersects every odd directed cycle of $D$. In the Directed Odd Cycle Transversal (DOCT) problem, the input consists of a digraph $D$ and an integer $k$. The objective is to determine whether there exists a directed odd cycle transversal of $D$ of size at most $k$. In this paper, we settle the parameteriz… ▽ More A directed odd cycle transversal of a directed graph (digraph) $D$ is a vertex set $S$ that intersects every odd directed cycle of $D$. In the Directed Odd Cycle Transversal (DOCT) problem, the input consists of a digraph $D$ and an integer $k$. The objective is to determine whether there exists a directed odd cycle transversal of $D$ of size at most $k$. In this paper, we settle the parameterized complexity of DOCT when parameterized by the solution size $k$ by showing that DOCT does not admit an algorithm with running time $f(k)n^{O(1)}$ unless FPT = W[1]. On the positive side, we give a factor $2$ fixed parameter tractable (FPT) approximation algorithm for the problem. More precisely, our algorithm takes as input $D$ and $k$, runs in time $2^{O(k^2)}n^{O(1)}$, and either concludes that $D$ does not have a directed odd cycle transversal of size at most $k$, or produces a solution of size at most $2k$. Finally, we provide evidence that there exists $ε> 0$ such that DOCT does not admit a factor $(1+ε)$ FPT-approximation algorithm. △ Less

Submitted 13 April, 2017; originally announced April 2017.

The Half-integral Erdös-Pósa Property for Non-null Cycles

Authors: Daniel Lokshtanov, M. S. Ramanujan, Saket Saurabh

Abstract: A Group Labeled Graph is a pair $(G,Λ)$ where $G$ is an oriented graph and $Λ$ is a mapping from the arcs of $G$ to elements of a group. A (not necessarily directed) cycle $C$ is called non-null if for any cyclic ordering of the arcs in $C$, the group element obtained by `adding' the labels on forward arcs and `subtracting' the labels on reverse arcs is not the identity element of the group. Non-n… ▽ More A Group Labeled Graph is a pair $(G,Λ)$ where $G$ is an oriented graph and $Λ$ is a mapping from the arcs of $G$ to elements of a group. A (not necessarily directed) cycle $C$ is called non-null if for any cyclic ordering of the arcs in $C$, the group element obtained by `adding' the labels on forward arcs and `subtracting' the labels on reverse arcs is not the identity element of the group. Non-null cycles in group labeled graphs generalize several well-known graph structures, including odd cycles. In this paper, we prove that non-null cycles on Group Labeled Graphs have the half-integral Erdös-Pósa property. That is, there is a function $f:{\mathbb N}\to {\mathbb N}$ such that for any $k\in {\mathbb N}$, any group labeled graph $(G,Λ)$ has a set of $k$ non-null cycles such that each vertex of $G$ appears in at most two of these cycles or there is a set of at most $f(k)$ vertices that intersects every non-null cycle. Since it is known that non-null cycles do not have the integeral Erdös-Pósa property in general, a half-integral Erdös-Pósa result is the best one could hope for. △ Less

Submitted 8 March, 2017; originally announced March 2017.

On finding highly connected spanning subgraphs

Authors: Manu Basavaraju, Pranabendu Misra, M. S. Ramanujan, Saket Saurabh

Abstract: In the Survivable Network Design Problem (SNDP), the input is an edge-weighted (di)graph $G$ and an integer $r_{uv}$ for every pair of vertices $u,v\in V(G)$. The objective is to construct a subgraph $H$ of minimum weight which contains $r_{uv}$ edge-disjoint (or node-disjoint) $u$-$v$ paths. This is a fundamental problem in combinatorial optimization that captures numerous well-studied problems i… ▽ More In the Survivable Network Design Problem (SNDP), the input is an edge-weighted (di)graph $G$ and an integer $r_{uv}$ for every pair of vertices $u,v\in V(G)$. The objective is to construct a subgraph $H$ of minimum weight which contains $r_{uv}$ edge-disjoint (or node-disjoint) $u$-$v$ paths. This is a fundamental problem in combinatorial optimization that captures numerous well-studied problems in graph theory and graph algorithms. In this paper, we consider the version of the problem where we are given a $λ$-edge connected (di)graph $G$ with a non-negative weight function $w$ on the edges and an integer $k$, and the objective is to find a minimum weight spanning subgraph $H$ that is also $λ$-edge connected, and has at least $k$ fewer edges than $G$. In other words, we are asked to compute a maximum weight subset of edges, of cardinality up to $k$, which may be safely deleted from $G$. Motivated by this question, we investigate the connectivity properties of $λ$-edge connected (di)graphs and obtain algorithmically significant structural results. We demonstrate the importance of our structural results by presenting an algorithm running in time $2^{O(k \log k)} |V(G)|^{O(1)}$ for $λ$-ECS, thus proving its fixed-parameter tractability. We follow up on this result and obtain the {\em first polynomial compression} for $λ$-ECS on unweighted graphs. As a consequence, we also obtain the first fixed parameter tractable algorithm, and a polynomial kernel for a parameterized version of the classic Mininum Equivalent Graph problem. We believe that our structural results are of independent interest and will play a crucial role in the design of algorithms for connectivity-constrained problems in general and the SNDP problem in particular. △ Less

Submitted 11 January, 2017; originally announced January 2017.

Backdoors to Tractable Valued CSP

Authors: Robert Ganian, M. S. Ramanujan, Stefan Szeider

Abstract: We extend the notion of a strong backdoor from the CSP setting to the Valued CSP setting (VCSP, for short). This provides a means for augmenting a class of tractable VCSP instances to instances that are outside the class but of small distance to the class, where the distance is measured in terms of the size of a smallest backdoor. We establish that VCSP is fixed-parameter tractable when parameteri… ▽ More We extend the notion of a strong backdoor from the CSP setting to the Valued CSP setting (VCSP, for short). This provides a means for augmenting a class of tractable VCSP instances to instances that are outside the class but of small distance to the class, where the distance is measured in terms of the size of a smallest backdoor. We establish that VCSP is fixed-parameter tractable when parameterized by the size of a smallest backdoor into every tractable class of VCSP instances characterized by a (possibly infinite) tractable valued constraint language of finite arity and finite domain. We further extend this fixed-parameter tractability result to so-called "scattered classes" of VCSP instances where each connected component may belong to a different tractable class. △ Less

Submitted 17 December, 2016; originally announced December 2016.

Comments: Accepted to CP 2016

Combining Treewidth and Backdoors for CSP

Authors: Robert Ganian, M. S. Ramanujan, Stefan Szeider

Abstract: We show that CSP is fixed-parameter tractable when parameterized by the treewidth of a backdoor into any tractable CSP problem over a finite constraint language. This result combines the two prominent approaches for achieving tractability for CSP: (i) by structural restrictions on the interaction between the variables and the constraints and (ii) by language restrictions on the relations that can… ▽ More We show that CSP is fixed-parameter tractable when parameterized by the treewidth of a backdoor into any tractable CSP problem over a finite constraint language. This result combines the two prominent approaches for achieving tractability for CSP: (i) by structural restrictions on the interaction between the variables and the constraints and (ii) by language restrictions on the relations that can be used inside the constraints. Apart from defining the notion of backdoor-treewidth and showing how backdoors of small treewidth can be used to efficiently solve CSP, our main technical contribution is a fixed-parameter algorithm that finds a backdoor of small treewidth. △ Less

Submitted 11 October, 2016; originally announced October 2016.

A Linear Time Parameterized Algorithm for Directed Feedback Vertex Set

Authors: Daniel Lokshtanov, M. S. Ramanujan, Saket Saurabh

Abstract: In the Directed Feedback Vertex Set (DFVS) problem, the input is a directed graph $D$ on $n$ vertices and $m$ edges, and an integer $k$. The objective is to determine whether there exists a set of at most $k$ vertices intersecting every directed cycle of $D$. Whether or not DFVS admits a fixed parameter tractable (FPT) algorithm was considered the most important open problem in parameterized compl… ▽ More In the Directed Feedback Vertex Set (DFVS) problem, the input is a directed graph $D$ on $n$ vertices and $m$ edges, and an integer $k$. The objective is to determine whether there exists a set of at most $k$ vertices intersecting every directed cycle of $D$. Whether or not DFVS admits a fixed parameter tractable (FPT) algorithm was considered the most important open problem in parameterized complexity until Chen, Liu, Lu, O'Sullivan and Razgon [JACM 2008] answered the question in the affirmative. They gave an algorithm for the problem with running time $O(k!4^kk^4nm)$. Since then, no faster algorithm for the problem has been found. In this paper, we give an algorithm for DFVS with running time $O(k!4^kk^5(n+m))$. Our algorithm is the first algorithm for DFVS with linear dependence on input size. Furthermore, the asymptotic dependence of the running time of our algorithm on the parameter $k$ matches up to a factor $k$ the algorithm of Chen, Liu, Lu, O'Sullivan and Razgon. On the way to designing our algorithm for DFVS, we give a general methodology to shave off a factor of $n$ from iterative-compression based algorithms for a few other well-studied covering problems in parameterized complexity. We demonstrate the applicability of this technique by speeding up by a factor of $n$, the current best FPT algorithms for Multicut [STOC 2011, SICOMP 2014] and Directed Subset Feedback Vertex Set [ICALP 2012, TALG 2014]. △ Less

Submitted 14 September, 2016; originally announced September 2016.

On the Optimality of Pseudo-polynomial Algorithms for Integer Programming

Authors: Fedor V. Fomin, Fahad Panolan, M. S. Ramanujan, Saket Saurabh

Abstract: In the classic Integer Programming (IP) problem, the objective is to decide whether, for a given $m \times n$ matrix $A$ and an $m$-vector $b=(b_1,\dots, b_m)$, there is a non-negative integer $n$-vector $x$ such that $Ax=b$. Solving (IP) is an important step in numerous algorithms and it is important to obtain an understanding of the precise complexity of this problem as a function of natural par… ▽ More In the classic Integer Programming (IP) problem, the objective is to decide whether, for a given $m \times n$ matrix $A$ and an $m$-vector $b=(b_1,\dots, b_m)$, there is a non-negative integer $n$-vector $x$ such that $Ax=b$. Solving (IP) is an important step in numerous algorithms and it is important to obtain an understanding of the precise complexity of this problem as a function of natural parameters of the input. The classic pseudo-polynomial time algorithm of Papadimitriou [J. ACM 1981] for instances of (IP) with a constant number of constraints was only recently improved upon by Eisenbrand and Weismantel [SODA 2018] and Jansen and Rohwedder [ArXiv 2018]. We continue this line of work and show that under the Exponential Time Hypothesis (ETH), the algorithm of Jansen and Rohwedder is nearly optimal. We also show that when the matrix $A$ is assumed to be non-negative, a component of Papadimitriou's original algorithm is already nearly optimal under ETH. This motivates us to pick up the line of research initiated by Cunningham and Geelen [IPCO 2007] who studied the complexity of solving (IP) with non-negative matrices in which the number of constraints may be unbounded, but the branch-width of the column-matroid corresponding to the constraint matrix is a constant. We prove a lower bound on the complexity of solving (IP) for such instances and obtain optimal results with respect to a closely related parameter, path-width. Specifically, we prove matching upper and lower bounds for (IP) when the path-width of the corresponding column-matroid is a constant. △ Less

Submitted 17 July, 2018; v1 submitted 18 July, 2016; originally announced July 2016.

Comments: 29 pages, To appear in ESA 2018

A Linear Time Parameterized Algorithm for Node Unique Label Cover

Authors: Daniel Lokshtanov, M. S. Ramanujan, Saket Saurabh

Abstract: The optimization version of the Unique Label Cover problem is at the heart of the Unique Games Conjecture which has played an important role in the proof of several tight inapproximability results. In recent years, this problem has been also studied extensively from the point of view of parameterized complexity. Cygan et al. [FOCS 2012] proved that this problem is fixed-parameter tractable (FPT) a… ▽ More The optimization version of the Unique Label Cover problem is at the heart of the Unique Games Conjecture which has played an important role in the proof of several tight inapproximability results. In recent years, this problem has been also studied extensively from the point of view of parameterized complexity. Cygan et al. [FOCS 2012] proved that this problem is fixed-parameter tractable (FPT) and Wahlström [SODA 2014] gave an FPT algorithm with an improved parameter dependence. Subsequently, Iwata, Wahlström and Yoshida [2014] proved that the edge version of Unique Label Cover can be solved in linear FPT-time. That is, there is an FPT algorithm whose dependence on the input-size is linear. However, such an algorithm for the node version of the problem was left as an open problem. In this paper, we resolve this question by presenting the first linear-time FPT algorithm for Node Unique Label Cover. △ Less

Submitted 29 April, 2016; originally announced April 2016.

Lossy Kernelization

Authors: Daniel Lokshtanov, Fahad Panolan, M. S. Ramanujan, Saket Saurabh

Abstract: In this paper we propose a new framework for analyzing the performance of preprocessing algorithms. Our framework builds on the notion of kernelization from parameterized complexity. However, as opposed to the original notion of kernelization, our definitions combine well with approximation algorithms and heuristics. The key new definition is that of a polynomial size $α$-approximate kernel. Loose… ▽ More In this paper we propose a new framework for analyzing the performance of preprocessing algorithms. Our framework builds on the notion of kernelization from parameterized complexity. However, as opposed to the original notion of kernelization, our definitions combine well with approximation algorithms and heuristics. The key new definition is that of a polynomial size $α$-approximate kernel. Loosely speaking, a polynomial size $α$-approximate kernel is a polynomial time pre-processing algorithm that takes as input an instance $(I,k)$ to a parameterized problem, and outputs another instance $(I',k')$ to the same problem, such that $|I'|+k' \leq k^{O(1)}$. Additionally, for every $c \geq 1$, a $c$-approximate solution $s'$ to the pre-processed instance $(I',k')$ can be turned in polynomial time into a $(c \cdot α)$-approximate solution $s$ to the original instance $(I,k)$. Our main technical contribution are $α$-approximate kernels of polynomial size for three problems, namely Connected Vertex Cover, Disjoint Cycle Packing and Disjoint Factors. These problems are known not to admit any polynomial size kernels unless $NP \subseteq coNP/poly$. Our approximate kernels simultaneously beat both the lower bounds on the (normal) kernel size, and the hardness of approximation lower bounds for all three problems. On the negative side we prove that Longest Path parameterized by the length of the path and Set Cover parameterized by the universe size do not admit even an $α$-approximate kernel of polynomial size, for any $α\geq 1$, unless $NP \subseteq coNP/poly$. In order to prove this lower bound we need to combine in a non-trivial way the techniques used for showing kernelization lower bounds with the methods for showing hardness of approximation △ Less

Submitted 4 November, 2016; v1 submitted 14 April, 2016; originally announced April 2016.

Comments: 58 pages. Version 2 contain new results: PSAKS for Cycle Packing and approximate kernel lower bounds for Set Cover and Hitting Set parameterized by universe size

Strong Backdoors for Linear Temporal Logic

Authors: Arne Meier, Sebastian Ordyniak, M. S. Ramanujan, Irena Schindler

Abstract: In the present paper we introduce the notion of strong backdoors into the field of temporal logic for the CNF-fragment of linear temporal logic introduced by Fisher. We study the parameterised complexity of the satisfiability problem parameterised by the size of the backdoor. We distinguish between backdoor detection and evaluation of backdoors into the fragments of horn and krom formulas. Here we… ▽ More In the present paper we introduce the notion of strong backdoors into the field of temporal logic for the CNF-fragment of linear temporal logic introduced by Fisher. We study the parameterised complexity of the satisfiability problem parameterised by the size of the backdoor. We distinguish between backdoor detection and evaluation of backdoors into the fragments of horn and krom formulas. Here we classify the operator fragments of globally-operators for past or future, and the combination of both. Detection is shown to be in FPT whereas the complexity of evaluation behaves different. We show that for krom formulas the problem is paraNP-complete. For horn formulas the complexity is shown to be either fixed parameter tractable or paraNP-complete depending on the considered operator fragment. △ Less

Submitted 16 February, 2016; originally announced February 2016.

Metric Dimension of Bounded Tree-length Graphs

Authors: Rémy Belmonte, Fedor V. Fomin, Petr A. Golovach, M. S. Ramanujan

Abstract: The notion of resolving sets in a graph was introduced by Slater (1975) and Harary and Melter (1976) as a way of uniquely identifying every vertex in a graph. A set of vertices in a graph is a resolving set if for any pair of vertices x and y there is a vertex in the set which has distinct distances to x and y. A smallest resolving set in a graph is called a metric basis and its size, the metric d… ▽ More The notion of resolving sets in a graph was introduced by Slater (1975) and Harary and Melter (1976) as a way of uniquely identifying every vertex in a graph. A set of vertices in a graph is a resolving set if for any pair of vertices x and y there is a vertex in the set which has distinct distances to x and y. A smallest resolving set in a graph is called a metric basis and its size, the metric dimension of the graph. The problem of computing the metric dimension of a graph is a well-known NP-hard problem and while it was known to be polynomial time solvable on trees, it is only recently that efforts have been made to understand its computational complexity on various restricted graph classes. In recent work, Foucaud et al. (2015) showed that this problem is NP-complete even on interval graphs. They complemented this result by also showing that it is fixed-parameter tractable (FPT) parameterized by the metric dimension of the graph. In this work, we show that this FPT result can in fact be extended to all graphs of bounded tree-length. This includes well-known classes like chordal graphs, AT-free graphs and permutation graphs. We also show that this problem is FPT parameterized by the modular-width of the input graph. △ Less

Submitted 8 February, 2016; originally announced February 2016.

A Parameterized Algorithm for Mixed Cut

Authors: Ashutosh Rai, M. S. Ramanujan, Saket Saurabh

Abstract: The classical Menger's theorem states that in any undirected (or directed) graph $G$, given a pair of vertices $s$ and $t$, the maximum number of vertex (edge) disjoint paths is equal to the minimum number of vertices (edges) needed to disconnect from $s$ and $t$. This min-max result can be turned into a polynomial time algorithm to find the maximum number of vertex (edge) disjoint paths as well a… ▽ More The classical Menger's theorem states that in any undirected (or directed) graph $G$, given a pair of vertices $s$ and $t$, the maximum number of vertex (edge) disjoint paths is equal to the minimum number of vertices (edges) needed to disconnect from $s$ and $t$. This min-max result can be turned into a polynomial time algorithm to find the maximum number of vertex (edge) disjoint paths as well as the minimum number of vertices (edges) needed to disconnect $s$ from $t$. In this paper we study a mixed version of this problem, called Mixed-Cut, where we are given an undirected graph $G$, vertices $s$ and $t$, positive integers $k$ and $l$ and the objective is to test whether there exist a $k$ sized vertex set $S \subseteq V(G)$ and an $l$ sized edge set $F \subseteq E(G)$ such that deletion of $S$ and $F$ from $G$ disconnects from $s$ and $t$. We start with a small observation that this problem is NP-complete and then study this problem, in fact a much stronger generalization of this, in the realm of parameterized complexity. In particular we study the Mixed-Multiway Cut-Uncut problem where along with a set of terminals $T$, we are also given an equivalence relation $\mathcal{R}$ on $T$, and the question is whether we can delete at most $k$ vertices and at most $l$ edges such that connectivity of the terminals in the resulting graph respects $\mathcal{R}$. Our main results is a fixed parameter algorithm for Mixed-Multiway Cut-Uncut using the method of recursive understanding introduced by Chitnis et al. (FOCS 2012). △ Less

Submitted 18 September, 2015; originally announced September 2015.

Comments: 16 pages. arXiv admin note: substantial text overlap with arXiv:1207.4079 by other authors

Discovering Archipelagos of Tractability for Constraint Satisfaction and Counting

Authors: Robert Ganian, M. S. Ramanujan, Stefan Szeider

Abstract: The Constraint Satisfaction Problem (CSP) is a central and generic computational problem which provides a common framework for many theoretical and practical applications. A central line of research is concerned with the identification of classes of instances for which CSP can be solved in polynomial time; such classes are often called "islands of tractability." A prominent way of defining islands… ▽ More The Constraint Satisfaction Problem (CSP) is a central and generic computational problem which provides a common framework for many theoretical and practical applications. A central line of research is concerned with the identification of classes of instances for which CSP can be solved in polynomial time; such classes are often called "islands of tractability." A prominent way of defining islands of tractability for CSP is to restrict the relations that may occur in the constraints to a fixed set, called a constraint language, whereas a constraint language is conservative if it contains all unary relations. This paper addresses the general limit of the mentioned tractability results for CSP and #CSP, that they only apply to instances where all constraints belong to a single tractable language (in general, the union of two tractable languages isn't tractable). We show that we can overcome this limitation as long as we keep some control of how constraints over the various considered tractable languages interact with each other. For this purpose we utilize the notion of a \emph{strong backdoor} of a CSP instance, as introduced by Williams et al. (IJCAI 2003), which is a set of variables that when instantiated moves the instance to an island of tractability, i.e., to a tractable class of instances. In this paper, we consider strong backdoors into \emph{scattered classes}, consisting of CSP instances where each connected component belongs entirely to some class from a list of tractable classes. Our main result is an algorithm that, given a CSP instance with $n$ variables, finds in time $f(k)n^{O(1)}$ a strong backdoor into a scattered class (associated with a list of finite conservative constraint languages) of size $k$ or correctly decides that there isn't such a backdoor. △ Less

Submitted 20 July, 2015; v1 submitted 9 July, 2015; originally announced July 2015.

FO Model Checking on Posets of Bounded Width

Authors: Jakub Gajarský, Petr Hliněný, Daniel Lokshtanov, Jan Obdržálek, Sebastian Ordyniak, M. S. Ramanujan, Saket Saurabh

Abstract: Over the past two decades the main focus of research into first-order (FO) model checking algorithms have been sparse relational structures-culminating in the FPT-algorithm by Grohe, Kreutzer and Siebertz for FO model checking of nowhere dense classes of graphs [STOC'14], with dense structures starting to attract attention only recently. Bova, Ganian and Szeider [LICS'14] initiated the study of th… ▽ More Over the past two decades the main focus of research into first-order (FO) model checking algorithms have been sparse relational structures-culminating in the FPT-algorithm by Grohe, Kreutzer and Siebertz for FO model checking of nowhere dense classes of graphs [STOC'14], with dense structures starting to attract attention only recently. Bova, Ganian and Szeider [LICS'14] initiated the study of the complexity of FO model checking on partially ordered sets (posets). Bova, Ganian and Szeider showed that model checking existential FO logic is fixed-parameter tractable (FPT) on posets of bounded width, where the width of a poset is the size of the largest antichain in the poset. The existence of an FPT algorithm for general FO model checking on posets of bounded width, however, remained open. We resolve this question in the positive by giving an algorithm that takes as its input an $n$-element poset $\cal P$ of width $w$ and an FO logic formula $φ$, and determines whether $φ$ holds on $\cal P$ in time $f(φ,w)\cdot n^2$. △ Less

Submitted 29 May, 2015; v1 submitted 16 April, 2015; originally announced April 2015.

Comments: Minor correction, p.5, def. of τ_{s+1}: instead of an induced subdigraph of D_s, we use the appropriate relational structure formed from this subdigrph

Reconfiguration on sparse graphs

Authors: Daniel Lokshtanov, Amer E. Mouawad, Fahad Panolan, M. S. Ramanujan, Saket Saurabh

Abstract: A vertex-subset graph problem Q defines which subsets of the vertices of an input graph are feasible solutions. A reconfiguration variant of a vertex-subset problem asks, given two feasible solutions S and T of size k, whether it is possible to transform S into T by a sequence of vertex additions and deletions such that each intermediate set is also a feasible solution of size bounded by k. We stu… ▽ More A vertex-subset graph problem Q defines which subsets of the vertices of an input graph are feasible solutions. A reconfiguration variant of a vertex-subset problem asks, given two feasible solutions S and T of size k, whether it is possible to transform S into T by a sequence of vertex additions and deletions such that each intermediate set is also a feasible solution of size bounded by k. We study reconfiguration variants of two classical vertex-subset problems, namely Independent Set and Dominating Set. We denote the former by ISR and the latter by DSR. Both ISR and DSR are PSPACE-complete on graphs of bounded bandwidth and W[1]-hard parameterized by k on general graphs. We show that ISR is fixed-parameter tractable parameterized by k when the input graph is of bounded degeneracy or nowhere-dense. As a corollary, we answer positively an open question concerning the parameterized complexity of the problem on graphs of bounded treewidth. Moreover, our techniques generalize recent results showing that ISR is fixed-parameter tractable on planar graphs and graphs of bounded degree. For DSR, we show the problem fixed-parameter tractable parameterized by k when the input graph does not contain large bicliques, a class of graphs which includes graphs of bounded degeneracy and nowhere-dense graphs. △ Less

Submitted 17 February, 2015; originally announced February 2015.

Linear Time Parameterized Algorithms via Skew-Symmetric Multicuts

Authors: M. S. Ramanujan, Saket Saurabh

Abstract: A skew-symmetric graph $(D=(V,A),σ)$ is a directed graph $D$ with an involution $σ$ on the set of vertices and arcs. In this paper, we introduce a separation problem, $d$-Skew-Symmetric Multicut, where we are given a skew-symmetric graph $D$, a family of $\cal T$ of $d$-sized subsets of vertices and an integer $k$. The objective is to decide if there is a set $X\subseteq A$ of $k$ arcs such that e… ▽ More A skew-symmetric graph $(D=(V,A),σ)$ is a directed graph $D$ with an involution $σ$ on the set of vertices and arcs. In this paper, we introduce a separation problem, $d$-Skew-Symmetric Multicut, where we are given a skew-symmetric graph $D$, a family of $\cal T$ of $d$-sized subsets of vertices and an integer $k$. The objective is to decide if there is a set $X\subseteq A$ of $k$ arcs such that every set $J$ in the family has a vertex $v$ such that $v$ and $σ(v)$ are in different connected components of $D'=(V,A\setminus (X\cup σ(X))$. In this paper, we give an algorithm for this problem which runs in time $O((4d)^{k}(m+n+\ell))$, where $m$ is the number of arcs in the graph, $n$ the number of vertices and $\ell$ the length of the family given in the input. Using our algorithm, we show that Almost 2-SAT has an algorithm with running time $O(4^kk^4\ell)$ and we obtain algorithms for {\sc Odd Cycle Transversal} and {\sc Edge Bipartization} which run in time $O(4^kk^4(m+n))$ and $O(4^kk^5(m+n))$ respectively. This resolves an open problem posed by Reed, Smith and Vetta [Operations Research Letters, 2003] and improves upon the earlier almost linear time algorithm of Kawarabayashi and Reed [SODA, 2010]. We also show that Deletion q-Horn Backdoor Set Detection is a special case of 3-Skew-Symmetric Multicut, giving us an algorithm for Deletion q-Horn Backdoor Set Detection which runs in time $O(12^kk^5\ell)$. This gives the first fixed-parameter tractable algorithm for this problem answering a question posed in a paper by a superset of the authors [STACS, 2013]. Using this result, we get an algorithm for Satisfiability which runs in time $O(12^kk^5\ell)$ where $k$ is the size of the smallest q-Horn deletion backdoor set, with $\ell$ being the length of the input formula. △ Less

Submitted 28 April, 2013; originally announced April 2013.

Fixed-Parameter Tractability of Workflow Satisfiability in the Presence of Seniority Constraints

Authors: J. Crampton, R. Crowston, G. Gutin, M. Jones, M. S. Ramanujan

Abstract: The workflow satisfiability problem is concerned with determining whether it is possible to find an allocation of authorized users to the steps in a workflow in such a way that all constraints are satisfied. The problem is NP-hard in general, but is known to be fixed-parameter tractable for certain classes of constraints. The known results on fixed-parameter tractability rely on the symmetry (in s… ▽ More The workflow satisfiability problem is concerned with determining whether it is possible to find an allocation of authorized users to the steps in a workflow in such a way that all constraints are satisfied. The problem is NP-hard in general, but is known to be fixed-parameter tractable for certain classes of constraints. The known results on fixed-parameter tractability rely on the symmetry (in some sense) of the constraints. In this paper, we provide the first results that establish fixed-parameter tractability of the satisfiability problem when the constraints are asymmetric. In particular, we introduce the notion of seniority constraints, in which the execution of steps is determined, in part, by the relative seniority of the users that perform them. Our results require new techniques, which make use of tree decompositions of the graph of the binary relation defining the constraint. Finally, we establish a lower bound for the hardness of the workflow satisfiability problem. △ Less

Submitted 15 October, 2012; originally announced October 2012.

Parameterized Complexity of Directed Steiner Tree on Sparse Graphs

Authors: Mark Jones, Daniel Lokshtanov, M. S. Ramanujan, Saket Saurabh, Ondřej Suchý

Abstract: We study the parameterized complexity of the directed variant of the classical {\sc Steiner Tree} problem on various classes of directed sparse graphs. While the parameterized complexity of {\sc Steiner Tree} parameterized by the number of terminals is well understood, not much is known about the parameterization by the number of non-terminals in the solution tree. All that is known for this param… ▽ More We study the parameterized complexity of the directed variant of the classical {\sc Steiner Tree} problem on various classes of directed sparse graphs. While the parameterized complexity of {\sc Steiner Tree} parameterized by the number of terminals is well understood, not much is known about the parameterization by the number of non-terminals in the solution tree. All that is known for this parameterization is that both the directed and the undirected versions are W[2]-hard on general graphs, and hence unlikely to be fixed parameter tractable FPT. The undirected {\sc Steiner Tree} problem becomes FPT when restricted to sparse classes of graphs such as planar graphs, but the techniques used to show this result break down on directed planar graphs. In this article we precisely chart the tractability border for {\sc Directed Steiner Tree} (DST) on sparse graphs parameterized by the number of non-terminals in the solution tree. Specifically, we show that the problem is fixed parameter tractable on graphs excluding a topological minor, but becomes W[2]-hard on graphs of degeneracy 2. On the other hand we show that if the subgraph induced by the terminals is required to be acyclic then the problem becomes FPT on graphs of bounded degeneracy. We further show that our algorithm achieves the best possible running time dependence on the solution size and degeneracy of the input graph, under standard complexity theoretic assumptions. Using the ideas developed for DST, we also obtain improved algorithms for {\sc Dominating Set} on sparse undirected graphs. These algorithms are asymptotically optimal. △ Less

Submitted 30 September, 2012; originally announced October 2012.

Comments: 28

Faster Parameterized Algorithms using Linear Programming

Authors: Daniel Lokshtanov, N. S. Narayanaswamy, Venkatesh Raman, M. S. Ramanujan, Saket Saurabh

Abstract: We investigate the parameterized complexity of Vertex Cover parameterized by the difference between the size of the optimal solution and the value of the linear programming (LP) relaxation of the problem. By carefully analyzing the change in the LP value in the branching steps, we argue that combining previously known preprocessing rules with the most straightforward branching algorithm yields an… ▽ More We investigate the parameterized complexity of Vertex Cover parameterized by the difference between the size of the optimal solution and the value of the linear programming (LP) relaxation of the problem. By carefully analyzing the change in the LP value in the branching steps, we argue that combining previously known preprocessing rules with the most straightforward branching algorithm yields an $O^*((2.618)^k)$ algorithm for the problem. Here $k$ is the excess of the vertex cover size over the LP optimum, and we write $O^*(f(k))$ for a time complexity of the form $O(f(k)n^{O(1)})$, where $f (k)$ grows exponentially with $k$. We proceed to show that a more sophisticated branching algorithm achieves a runtime of $O^*(2.3146^k)$. Following this, using known and new reductions, we give $O^*(2.3146^k)$ algorithms for the parameterized versions of Above Guarantee Vertex Cover, Odd Cycle Transversal, Split Vertex Deletion and Almost 2-SAT, and an $O^*(1.5214^k)$ algorithm for Konig Vertex Deletion, Vertex Cover Param by OCT and Vertex Cover Param by KVD. These algorithms significantly improve the best known bounds for these problems. The most notable improvement is the new bound for Odd Cycle Transversal - this is the first algorithm which beats the dependence on $k$ of the seminal $O^*(3^k)$ algorithm of Reed, Smith and Vetta. Finally, using our algorithm, we obtain a kernel for the standard parameterization of Vertex Cover with at most $2k - c \log k$ vertices. Our kernel is simpler than previously known kernels achieving the same size bound. △ Less

Submitted 12 March, 2012; v1 submitted 5 March, 2012; originally announced March 2012.

Comments: A preliminary version of this paper appears in the proceedings of STACS 2012