Discrete Mathematics
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Showing new listings for Monday, 24 August 2026
- [1] arXiv:2608.21248 [pdf, html, other]
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Title: T-Robinson Spaces: Structure, Recognition, and Applications to Real DataSubjects: Discrete Mathematics (cs.DM); Data Structures and Algorithms (cs.DS); Combinatorics (math.CO)
We study \emph{$T$-Robinson spaces}, a tree-based generalization of Robinson spaces in which every path of a compatible tree induces a Robinson subspace. This framework extends the classical notion of Robinsonian representations from linear orderings to tree structures, allowing the modeling of hierarchical and branching data. We establish a complete combinatorial characterization of $T$-Robinson spaces by proving their equivalence with several graph- and hypergraph-theoretic properties. In particular, we show that a dissimilarity space is $T$-Robinson if and only if all its level graphs are dually chordal with a common compatible tree. Combined with the characterization of hypertrees established by Brucker~\cite{brucker2005hypertrees}, this yields the equivalent characterization in terms of the associated cluster, ball, and 2-ball hypergraphs being hypertrees. Building upon these structural results, we develop a recognition algorithm with complexity \(O(K n^{2})\), where \(K\) denotes the number of minimum spanning trees of the dissimilarity space, improving upon existing hypertree-based approaches whenever \(K\) remains moderate. We further introduce a quantitative measure of $T$-Robinson structure that evaluates the extent to which an arbitrary dissimilarity space admits a tree-like representation. Finally, we discuss applications to real-world datasets, illustrating how $T$-Robinson spaces provide an interpretable framework for analyzing and organizing relational data.
New submissions (showing 1 of 1 entries)
- [2] arXiv:2608.20739 (cross-list from math.CO) [pdf, html, other]
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Title: Laziness of Quantum Walks on GraphsSubjects: Combinatorics (math.CO); Discrete Mathematics (cs.DM); Quantum Physics (quant-ph)
The trace of the average mixing matrix of a quantum walk measures the "laziness" of the walk: the higher the trace, the more likely that the walker returns home in the long run. In this paper, we develop tools to study this graph invariant arising from Laplacian quantum walks.
It is known that the complete graph $K_n$ is the laziest connected graph on $n$ vertices. Using our machinery, we show that the star $S_n$ is the second laziest connected graph on $n$ vertices (and hence the laziest tree on $n$ vertices), the complete multipartite graph $K_{n-2,1,1}$ is the third laziest connected graph on $n$ vertices, and the double star $DS(n-3,1)$ is the second laziest tree on $n$ vertices. We also show that on the same number of vertices, more unbalanced double stars are lazier. - [3] arXiv:2608.20894 (cross-list from math.OC) [pdf, other]
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Title: Maximum Covering Network Design on Graphs with Low Connectivity: Dynamic Programming and Block-Cut TreesSubjects: Optimization and Control (math.OC); Discrete Mathematics (cs.DM); Data Structures and Algorithms (cs.DS)
Planning accessible public services such as health care, emergency response, and schools often requires not only choosing where to open facilities but also improving the network that connects people to them, for example upgrading flood-prone roads in vulnerable regions. Most location models, however, take the network as fixed and the budget as given. We study the Maximum Covering Network Design Problem, in which a single budget is shared between opening facilities and upgrading weak links to maximize the population within a target travel distance of an open facility. The problem is hard even on the simplest networks, and planners usually want to see how coverage grows with the budget, not a single plan.
We develop an exact dynamic-programming framework that exploits a property common to real road networks: their low connectivity, with many cut points whose removal disconnects the network. On trees, the recursion is self-contained: each state reduces to a few simple facility and upgrade choices that are fast to compute without a solver, giving predictable running times; for larger budgets and travel distances it outperforms solving the MILP formulation directly. On general low-connectivity networks, the framework decomposes the problem at the cut points and embeds a given MILP formulation to solve the resulting pieces, coordinating them through coverage conditions at the interfaces. This lets us compare a formulation on its own against the same formulation inside the framework: across 306 test cases the framework matches or outperforms direct solving on more than 80% of instances. Because it evaluates all budget levels in a single run, it also yields the full coverage-versus-budget curve at no extra cost, whereas direct solving must split its time across individual budgets.
Cross submissions (showing 2 of 2 entries)
- [4] arXiv:2310.15152 (replaced) [pdf, html, other]
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Title: Sampling Balanced Forests of Grids in Polynomial TimeSubjects: Discrete Mathematics (cs.DM); Data Structures and Algorithms (cs.DS); Combinatorics (math.CO)
We prove that a polynomial fraction of the set of $k$-component forests in the $m \times n$ grid graph have equal numbers of vertices in each component, for any constant $k$. This resolves a conjecture of Charikar, Liu, Liu, and Vuong, and establishes the first provably polynomial-time algorithm for (exactly or approximately) sampling balanced grid graph partitions according to the spanning tree distribution, which weights each $k$-partition according to the product, across its $k$ pieces, of the number of spanning trees of each piece. Our result follows from a careful analysis of the probability a uniformly random spanning tree of the grid can be cut into balanced pieces.
Beyond grids, we show that for a broad family of lattice-like graphs, we achieve balance up to any multiplicative $(1 \pm \varepsilon)$ constant with constant probability, and up to an additive constant with polynomial probability. More generally, we show that, with constant probability, components derived from uniform spanning trees can approximate any given partition of a planar region specified by Jordan curves. These results imply polynomial time algorithms for sampling approximately balanced tree-weighted partitions for lattice-like graphs.
Our results have applications to understanding political districtings, where there is an underlying graph of indivisible geographic units that must be partitioned into $k$ population-balanced connected subgraphs. In this setting, tree-weighted partitions have interesting geometric properties, and this has stimulated significant effort to develop methods to sample them. - [5] arXiv:2404.19453 (replaced) [pdf, html, other]
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Title: Structural Parameters for Dense Temporal GraphsComments: 27 pages, 2 figuresSubjects: Discrete Mathematics (cs.DM); Combinatorics (math.CO)
Temporal graphs provide a useful model for many real-world networks. Unfortunately the majority of algorithmic problems we might consider on such graphs are intractable. There has been recent progress in defining structural parameters which describe tractable cases by simultaneously restricting the underlying structure and the times at which edges appear in the graph. These all rely on the temporal graph being sparse in some sense. We introduce temporal analogues of three increasingly restrictive static graph parameters -- cliquewidth, modular-width and neighbourhood diversity -- which take small values for highly structured temporal graphs, even if a large number of edges are active at each timestep. The computational problems solvable efficiently when the temporal cliquewidth of the input graph is bounded form a subset of those solvable efficiently when the temporal modular-width is bounded, which is in turn a subset of problems efficiently solvable when the temporal neighbourhood diversity is bounded. By considering specific temporal graph problems, we demonstrate that (up to standard complexity theoretic assumptions) these inclusions are strict.
- [6] arXiv:2507.00728 (replaced) [pdf, html, other]
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Title: Temporal Orienteering with Changing Fuel CostsComments: 21 pages, 2 figuresSubjects: Discrete Mathematics (cs.DM)
The problem Orienteering asks whether there exists a walk which visits a number of sites without exceeding some fuel budget. In the variant of the problem we consider, the cost of each edge in the walk is dependent on the time we depart one endpoint and the time we arrive at the other endpoint. This mirrors applications such as travel between orbiting objects where fuel costs are dependent on both the departure time and the length of time spent travelling. In defining this problem, we introduce a natural generalisation of the standard notion of temporal graphs: the pair consisting of the graph of the sites and a cost function, in which costs as well as shortest travel times between pairs of objects change over time. We believe this model is likely to be of independent interest. The problem of deciding whether a stated goal is feasible is easily seen to be NP-complete; we investigate three different ways to restrict the input which lead to efficient algorithms. These include the number of times an edge can be used, an analogue of vertex-interval-membership width, and the number of sites to be visited.
- [7] arXiv:2605.26852 (replaced) [pdf, html, other]
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Title: Recognizing Level-k-Based Phylogenetic Networks is NP-CompleteComments: 16 pages, 7 figures. v2: Abstract, Section 1, 3, 4, 6, and Acknowledgements editedSubjects: Populations and Evolution (q-bio.PE); Discrete Mathematics (cs.DM)
Phylogenetic networks generalize phylogenetic trees by representing reticulate evolution. Tree-based networks and their support trees have been extensively studied, but not all networks are tree-based. To measure how far such networks are from being tree-based, Suzuki and Hayamizu (2025) formulated the problem of finding the support network with minimum level of a given rooted almost-binary phylogenetic network. They conjectured that this problem is NP-hard and provided exponential-time algorithms. In this paper, we prove this conjecture by showing that, for every fixed integer $k \geq 1$, it is NP-complete to decide whether the minimum level is at most $k$.