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Graphs whose vertices of degree at least 2 lie in a triangle
Authors:
Vinicius L. do Forte,
Min Chih Lin,
Abilio Lucena,
Nelson Maculan,
Veronica A. Moyano,
Jayme L. Szwarcfiter
Abstract:
A pendant vertex is one of degree one and an isolated vertex has degree zero. A neighborhood star-free (NSF for short) graph is one in which every vertex is contained in a triangle except pendant vertices and isolated vertices. This class has been considered before for several contexts. In the present paper, we study the complexity of the dominating induced matching (DIM) problem and the perfect e…
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A pendant vertex is one of degree one and an isolated vertex has degree zero. A neighborhood star-free (NSF for short) graph is one in which every vertex is contained in a triangle except pendant vertices and isolated vertices. This class has been considered before for several contexts. In the present paper, we study the complexity of the dominating induced matching (DIM) problem and the perfect edge domination (PED) problem for NSF graphs. We prove the corresponding decision problems are NP-Complete for several of its subclasses. As an added value of this study, we have shown three connected variants of planar positive 1in3SAT are also NP-Complete. Since these variants are more basic in complexity theory context than many graph problems, these results can be useful to prove that other problems are NP-Complete.
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Submitted 7 April, 2024; v1 submitted 25 April, 2022;
originally announced April 2022.
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Cycle-based formulations in Distance Geometry
Authors:
Leo Liberti,
Gabriele Iommazzo,
Carlile Lavor,
Nelson Maculan
Abstract:
The distance geometry problem asks to find a realization of a given simple edge-weighted graph in a Euclidean space of given dimension K, where the edges are realized as straight segments of lengths equal (or as close as possible) to the edge weights. The problem is often modelled as a mathematical programming formulation involving decision variables that determine the position of the vertices in…
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The distance geometry problem asks to find a realization of a given simple edge-weighted graph in a Euclidean space of given dimension K, where the edges are realized as straight segments of lengths equal (or as close as possible) to the edge weights. The problem is often modelled as a mathematical programming formulation involving decision variables that determine the position of the vertices in the given Euclidean space. Solution algorithms are generally constructed using local or global nonlinear optimization techniques. We present a new modelling technique for this problem where, instead of deciding vertex positions, formulations decide the length of the segments representing the edges in each cycle in the graph, projected in every dimension. We propose an exact formulation and a relaxation based on a Eulerian cycle. We then compare computational results from protein conformation instances obtained with stochastic global optimization techniques on the new cycle-based formulation and on the existing edge-based formulation. While edge-based formulations take less time to reach termination, cycle-based formulations are generally better on solution quality measures.
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Submitted 28 July, 2023; v1 submitted 20 June, 2020;
originally announced June 2020.
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New error measures and methods for realizing protein graphs from distance data
Authors:
Claudia D'Ambrosio,
Ky Vu,
Carlile Lavor,
Leo Liberti,
Nelson Maculan
Abstract:
The interval Distance Geometry Problem (iDGP) consists in finding a realization in $\mathbb{R}^K$ of a simple undirected graph $G=(V,E)$ with nonnegative intervals assigned to the edges in such a way that, for each edge, the Euclidean distance between the realization of the adjacent vertices is within the edge interval bounds. In this paper, we focus on the application to the conformation of prote…
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The interval Distance Geometry Problem (iDGP) consists in finding a realization in $\mathbb{R}^K$ of a simple undirected graph $G=(V,E)$ with nonnegative intervals assigned to the edges in such a way that, for each edge, the Euclidean distance between the realization of the adjacent vertices is within the edge interval bounds. In this paper, we focus on the application to the conformation of proteins in space, which is a basic step in determining protein function: given interval estimations of some of the inter-atomic distances, find their shape. Among different families of methods for accomplishing this task, we look at mathematical programming based methods, which are well suited for dealing with intervals. The basic question we want to answer is: what is the best such method for the problem? The most meaningful error measure for evaluating solution quality is the coordinate root mean square deviation. We first introduce a new error measure which addresses a particular feature of protein backbones, i.e. many partial reflections also yield acceptable backbones. We then present a set of new and existing quadratic and semidefinite programming formulations of this problem, and a set of new and existing methods for solving these formulations. Finally, we perform a computational evaluation of all the feasible solver$+$formulation combinations according to new and existing error measures, finding that the best methodology is a new heuristic method based on multiplicative weights updates.
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Submitted 4 July, 2016;
originally announced July 2016.
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Distance geometry approach for special graph coloring problems
Authors:
Rosiane de Freitas,
Bruno Dias,
Nelson Maculan,
Jayme Szwarcfiter
Abstract:
One of the most important combinatorial optimization problems is graph coloring. There are several variations of this problem involving additional constraints either on vertices or edges. They constitute models for real applications, such as channel assignment in mobile wireless networks. In this work, we consider some coloring problems involving distance constraints as weighted edges, modeling th…
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One of the most important combinatorial optimization problems is graph coloring. There are several variations of this problem involving additional constraints either on vertices or edges. They constitute models for real applications, such as channel assignment in mobile wireless networks. In this work, we consider some coloring problems involving distance constraints as weighted edges, modeling them as distance geometry problems. Thus, the vertices of the graph are considered as embedded on the real line and the coloring is treated as an assignment of positive integers to the vertices, while the distances correspond to line segments, where the goal is to find a feasible intersection of them. We formulate different such coloring problems and show feasibility conditions for some problems. We also propose implicit enumeration methods for some of the optimization problems based on branch-and-prune methods proposed for distance geometry problems in the literature. An empirical analysis was undertaken, considering equality and inequality constraints, uniform and arbitrary set of distances, and the performance of each variant of the method considering the handling and propagation of the set of distances involved.
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Submitted 15 June, 2016;
originally announced June 2016.
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A speed and departure time optimization algorithm for the Pollution-Routing Problem
Authors:
Raphael Kramer,
Nelson Maculan,
Anand Subramanian,
Thibaut Vidal
Abstract:
We propose a new speed and departure time optimization algorithm for the Pollution-Routing Problem (PRP), which runs in quadratic time and returns a certified optimal schedule. This algorithm is embedded into an iterated local search-based metaheuristic to achieve a combined speed, scheduling and routing optimization. The start of the working day is set as a decision variable for individual routes…
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We propose a new speed and departure time optimization algorithm for the Pollution-Routing Problem (PRP), which runs in quadratic time and returns a certified optimal schedule. This algorithm is embedded into an iterated local search-based metaheuristic to achieve a combined speed, scheduling and routing optimization. The start of the working day is set as a decision variable for individual routes, thus enabling a better assignment of human resources to required demands. Some routes that were evaluated as unprofitable can now appear as viable candidates later in the day, leading to a larger search space and further opportunities of distance optimization via better service consolidation. Extensive computational experiments on available PRP benchmark instances demonstrate the good performance of the algorithms. The flexible departure times from the depot contribute to reduce the operational costs by 8.36% on the considered instances.
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Submitted 31 March, 2018; v1 submitted 21 January, 2015;
originally announced January 2015.
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A Decomposition Algorithm for Nested Resource Allocation Problems
Authors:
Thibaut Vidal,
Patrick Jaillet,
Nelson Maculan
Abstract:
We propose an exact polynomial algorithm for a resource allocation problem with convex costs and constraints on partial sums of resource consumptions, in the presence of either continuous or integer variables. No assumption of strict convexity or differentiability is needed. The method solves a hierarchy of resource allocation subproblems, whose solutions are used to convert constraints on sums of…
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We propose an exact polynomial algorithm for a resource allocation problem with convex costs and constraints on partial sums of resource consumptions, in the presence of either continuous or integer variables. No assumption of strict convexity or differentiability is needed. The method solves a hierarchy of resource allocation subproblems, whose solutions are used to convert constraints on sums of resources into bounds for separate variables at higher levels. The resulting time complexity for the integer problem is $O(n \log m \log (B/n))$, and the complexity of obtaining an $ε$-approximate solution for the continuous case is $O(n \log m \log (B/ε))$, $n$ being the number of variables, $m$ the number of ascending constraints (such that $m < n$), $ε$ a desired precision, and $B$ the total resource. This algorithm attains the best-known complexity when $m = n$, and improves it when $\log m = o(\log n)$. Extensive experimental analyses are conducted with four recent algorithms on various continuous problems issued from theory and practice. The proposed method achieves a higher performance than previous algorithms, addressing all problems with up to one million variables in less than one minute on a modern computer.
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Submitted 26 April, 2014;
originally announced April 2014.
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Large neighborhoods with implicit customer selection for vehicle routing problems with profits
Authors:
Thibaut Vidal,
Nelson Maculan,
Puca Huachi Vaz Penna,
Luis Satoru Ochi
Abstract:
We consider several Vehicle Routing Problems (VRP) with profits, which seek to select a subset of customers, each one being associated with a profit, and to design service itineraries. When the sum of profits is maximized under distance constraints, the problem is usually called team orienteering problem. The capacitated profitable tour problem seeks to maximize profits minus travel costs under ca…
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We consider several Vehicle Routing Problems (VRP) with profits, which seek to select a subset of customers, each one being associated with a profit, and to design service itineraries. When the sum of profits is maximized under distance constraints, the problem is usually called team orienteering problem. The capacitated profitable tour problem seeks to maximize profits minus travel costs under capacity constraints. Finally, in the VRP with private fleet and common carrier, some customers can be delegated to an external carrier subject to a cost. Three families of combined decisions must be taken: customers selection, assignment to vehicles, and sequencing of deliveries for each route.
We propose a new neighborhood search for these problems which explores an exponential number of solutions in pseudo polynomial time. The search is conducted with standard VRP neighborhoods on an "exhaustive" solution representation, visiting all customers. Since visiting all customers is usually infeasible or sub-optimal, an efficient "Select" algorithm, based on resource constrained shortest paths, is repeatedly used on any new route to find the optimal subsequence of visits to customers. The good performance of these neighborhood structures is demonstrated by extensive computational experiments with a local search, an iterated local search and a hybrid genetic algorithm. Intriguingly, even a local-improvement method to the first local optimum of this neighborhood achieves an average gap of 0.09% on classic team orienteering benchmark instances, rivaling with the current state-of-the-art metaheuristics. Promising research avenues on hybridizations with more standard routing neighborhoods are also open.
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Submitted 26 July, 2014; v1 submitted 15 January, 2014;
originally announced January 2014.
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An Integer Programming Formulation Applied to Optimum Allocation in Multivariate Stratified Sampling
Authors:
Jose Andre de Moura Brito,
Gustavo Silva Semaan,
Pedro Luis do Nascimento Silva,
Nelson Maculan
Abstract:
The problem of optimal allocation of samples in surveys using a stratified sampling plan was first discussed by Neyman in 1934. Since then, many researchers have studied the problem of the sample allocation in multivariate surveys and several methods have been proposed. Basically, these methods are divided into two class: The first involves forming a weighted average of the stratum variances and f…
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The problem of optimal allocation of samples in surveys using a stratified sampling plan was first discussed by Neyman in 1934. Since then, many researchers have studied the problem of the sample allocation in multivariate surveys and several methods have been proposed. Basically, these methods are divided into two class: The first involves forming a weighted average of the stratum variances and finding the optimal allocation for the average variance. The second class is associated with methods that require that an acceptable coefficient of variation for each of the variables on which the allocation is to be done. Particularly, this paper proposes a new optimization approach to the second problem. This approach is based on an integer programming formulation. Several experiments showed that the proposed approach is efficient way to solve this problem, considering a comparison of this approach with the other approach from the literature.
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Submitted 24 September, 2013;
originally announced September 2013.
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An Exact Algorithm for the Stratification Problem with Proportional Allocation
Authors:
Jose Brito,
Mauricio Lila,
Flavio Montenegro,
Nelson Maculan
Abstract:
We report a new optimal resolution for the statistical stratification problem under proportional sampling allocation among strata. Consider a finite population of N units, a random sample of n units selected from this population and a number L of strata. Thus, we have to define which units belong to each stratum so as to minimize the variance of a total estimator for one desired variable of inte…
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We report a new optimal resolution for the statistical stratification problem under proportional sampling allocation among strata. Consider a finite population of N units, a random sample of n units selected from this population and a number L of strata. Thus, we have to define which units belong to each stratum so as to minimize the variance of a total estimator for one desired variable of interest in each stratum,and consequently reduce the overall variance for such quantity. In order to solve this problem, an exact algorithm based on the concept of minimal path in a graph is proposed and assessed. Computational results using real data from IBGE (Brazilian Central Statistical Office) are provided.
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Submitted 18 February, 2009;
originally announced February 2009.
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Acyclic orientations with path constraints
Authors:
Rosa M. V. Figueiredo,
Valmir C. Barbosa,
Nelson Maculan,
Cid C. Souza
Abstract:
Many well-known combinatorial optimization problems can be stated over the set of acyclic orientations of an undirected graph. For example, acyclic orientations with certain diameter constraints are closely related to the optimal solutions of the vertex coloring and frequency assignment problems. In this paper we introduce a linear programming formulation of acyclic orientations with path constr…
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Many well-known combinatorial optimization problems can be stated over the set of acyclic orientations of an undirected graph. For example, acyclic orientations with certain diameter constraints are closely related to the optimal solutions of the vertex coloring and frequency assignment problems. In this paper we introduce a linear programming formulation of acyclic orientations with path constraints, and discuss its use in the solution of the vertex coloring problem and some versions of the frequency assignment problem. A study of the polytope associated with the formulation is presented, including proofs of which constraints of the formulation are facet-defining and the introduction of new classes of valid inequalities.
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Submitted 21 October, 2005;
originally announced October 2005.