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cs.DM: explore 112 source-linked works published from 2019 to 2026, with original documents and citations.
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Sources: arxiv. Collection updated 2026-09-16. Counts describe this index, not the complete source archives.
We show that Partizan Arc Kayles (PArcK), a generalization of Domineering to graphs, is PSPACE-complete via a reduction from Positive CNF and with recently-discovered techniques for creating PArcK positions with high temperature. The reduction uses only red and blue edges.
Combinatorial optimization (CO) problems are often NP-hard and thus out of reach for exact algorithms, making them a tempting domain to apply machine learning methods. The highly structured constraints in these problems can hinder either optimization or sampling directly in the solution space. On the other hand, GFlowNets have recently emerged as a powerful machinery to efficiently sample from composite unnormalized densities sequentially and have the potential to amortize such solution-searching processes in CO, as well as generate diverse solution candidates. In this paper, we design Markov decision processes (MDPs) for different combinatorial problems and propose to train conditional GFlowNets to sample from the solution space. Efficient training techniques are also developed to benefit long-range credit assignment. Through extensive experiments on a variety of different CO tasks with synthetic and realistic data, we demonstrate that GFlowNet policies can efficiently find high-quality solutions. Our implementation is open-sourced at https://github.com/zdhNarsil/GFlowNet-CombOpt.
We study flow shop scheduling with stochastic reentry, where jobs must complete multiple passes through the entire shop, and the number of passes that a job requires for completion is drawn from a discrete probability distribution. The goal is to find policies that minimize performance measures in expectation. Our main contribution is a reduction to a stochastic scheduling problem on identical parallel machines augmented by machine arrivals. This reduction preserves objective values and enables the transfer of structural results and performance guarantees from the auxiliary problems to the reentrant flow shop setting. We demonstrate the usefulness of this reduction by proving the optimality of simple priority policies for minimizing the makespan and the total completion time in expectation under geometric and, more generally, monotone hazard rate distributions. For minimizing the total weighted completion time, we derive an approximation guarantee for a simple priority policy that depends only on the squared coefficient of variation of the underlying distributions. Our results constitute the first optimality and approximation guarantees for flow shops with stochastic reentry and demonstrate that established scheduling policies naturally extend to this setting through the proposed reduction.
Matching Vector families (MVFs) are defined by two ordered lists of vectors in $\mathbb{Z}_m^n$ whose inner products satisfy specific residue patterns modulo an integer $m$. Most famously, restricted MVFs are used to construct the best-known constant-query Locally Decodable codes (LDCs). We prove an upper bound of $2^{O\left(\sqrt{n\log n \log m}\right)}$ on the size of $3$-restricted MVFs in $\mathbb{Z}_m^n$ for $m \leq \sqrt{n}$, substantially improving on the previous best bound of $2^{O(n/\log n)}$ by Bhowmick, Dvir and Lovett (STOC'13, SICOMP'14). Our proof relies on a new polynomial method argument that controls collisions in sumsets of matching vectors.
We study the combinatorial (red-blue) discrepancy of finite point sets with respect to hyperplanes and, more generally, bounded-complexity affine algebraic sets. We prove that every $n$-point set in a real Euclidean space admits a red-blue coloring for which every affine algebraic set of dimension at most $D$ and degree at most $k$ has discrepancy at most $n^{\frac12-\frac{1}{2(D+1)}-\varepsilon}$ for some $\varepsilon=\varepsilon(D,k)>0$. This gives a polynomial improvement over the straightforward VC-dimension bound $\tilde O(n^{\frac12-\frac{1}{2(D+1)}})$. In the opposite direction, we construct $n$-point sets in $\mathbb R^d$ whose discrepancy with respect to hyperplanes is $\tildeΩ(n^{\frac12-\frac{1}{d+1}}),$ extending the point-line discrepancy lower bound of Chazelle and Lvov. We present further applications of our methods in communication complexity, concerning separation between randomized communication cost and deterministic communication cost with access to equality oracle.
A conjecture of Komlós states that the combinatorial discrepancy of any matrix $A\in\mathbb R^{m\times n}$ whose columns have Euclidean norm at most one is bounded by a universal constant. We prove that the combinatorial discrepancy of every such matrix is at most $O((\log n)^{1/4}(\log\log n)^{7/4})$. This is the first asymptotic improvement over the $O(\sqrt{\log n})$ bound established by Banaszczyk [Banaszczyk, Random Struct.\ Algorithms, 1998], and it refutes a conjecture of Hajela [Hajela, European J.\ Combin., 1988] that a lower bound of order $Ω(\sqrt{\log n})$ should hold.
A graphical design is a subset of vertices of a graph, along with a weight for each chosen vertex, that can perfectly average chosen subspaces of functions on the graph. A design is uniformly weighted if all the weights are equal, and several well-known combinatorial objects such as orthogonal arrays, combinatorial block designs and t-wise permutations are uniformly weighted graphical designs. While one might expect to see uniformly weighted designs in structured graphs, they do not always exist. In this paper we characterize the existence of uniformly weighted graphical designs, and use our result to provide several families of graphs that have, and do not have, such designs. Our results offer a polyhedral view of the structures that control the existence and cardinalities of these designs. In particular, we characterize all uniformly weighted designs of threshold graphs, and provide a geometric proof of the duality of linear codes and linear orthogonal arrays. We also provide a novel construction for graphs whose Laplacian characteristic polynomials are almost irreducible, to produce families without uniformly weighted designs.
We study path-width and the closely related parameter line-width in countably infinite graphs. Our first result characterises the graphs of finite path-width: they are the graphs that do not have infinitely many vertices of infinite degree, do not have infinitely many pairwise disjoint infinite paths, and contain no subdivision of some finite tree of maximum degree 3. We then investigate universality under the subgraph relation for graphs of bounded path-width or line-width. In particular, we prove that there exists a universal graph with line-width $\mathcal{O}(k^2)$ for the class of graphs with line-width at most $k$. In contrast, we show that no graph of finite path-width is universal for the class of locally finite graphs with path-width $1$. Finally, we show that for each $k\geq 2$, every universal graph for the class of graphs with path-width at most $k$ has line-width at least $k + 1$.
We study the linear recurrence order of the Potts model partition function on thermal 2D grid graphs. By restricting the transfer matrix (TM) to the real coupling axis, the physical operator maintains diagonalizability under the Spectral Theorem. We show that this thermal regularization allows the Krylov subspace to saturate the unconstrained planar state capacity, locking the recurrence order to the Dyck path up to reversal (OEIS A007123) for $q \ge 4$. Furthermore, the recurrence order collapses to height-restricted Dyck paths up to reversal (OEIS A001998) for $q=3$ due to finite-index Jones-Wenzl projections, and to the zero-magnetization conservation sector (OEIS A001405) for $q=2$. This framework bridges graph-theoretic combinatorics with the representation theory of physical loop gas models.
Phylogenetic networks model evolutionary histories that involve reticulate events, but their structural complexity makes them difficult to interpret. Extracting their simple substructures both clarifies the evolutionary pathways and quantifies the complexity of the networks themselves. For a given rooted almost-binary phylogenetic network, the Level Minimization problem asks for a spanning subgraph that has the same root and leaf-set and whose level is minimum, i.e., which is as close to a tree as possible. Networks for which the minimum level is zero are known as tree-based networks and can be recognized in linear time. However, Level Minimization is NP-hard in general. State-of-the-art algorithms rely on exhaustive searches of the solution spaces and hence apply only to networks of limited size. In this paper, we propose two methods for Level Minimization using integer linear programming: an exact formulation for finding such a subgraph of level at most one, and a heuristic formulation for the general case. Computational experiments confirmed the practicality of both formulations. An application to ancestral recombination graphs suggests that the minimum level provides an alternative measure of the topological complexity of an inferred network.
An $st$-orientation of an undirected graph $G$ is an acyclic digraph with a single source $s$ and a single sink $t$ that can be obtained from $G$ by assigning a direction to each edge. The classical problem of deciding if an undirected graph $G$ has an $st$-orientation can be solved efficiently. On the other hand, deciding if an $st$-orientation of $G$ exists that does not have any transitive arc is NP-complete, even if each vertex of $G$ has degree at most four. Here we show that this last decision problem remains NP-complete if $G$ is cubic, which settles an open question by Binucci et al. (2025). We obtain NP-completeness for two variants of the problem: (i) $s$ and $t$ are fixed and given as part of the input and (ii) $s$ and $t$ can be chosen freely. We then use these results to investigate the computational complexity of a problem that arises in computational evolution. Specifically, we show that the problem of deciding if an unrooted binary phylogenetic network has an orientation as a rooted binary phylogenetic network without any shortcuts (the analog of a transitive arcs in phylogenetics) is NP-complete. Our results connect the two (mostly) distinct research areas of orienting undirected graphs and orienting unrooted phylogenetic networks.
For a graph $G$, denote by $a(G)$ the number of vertices in the largest induced forest in $G$. The Albertson-Berman conjecture, which has been open since 1979, states that $a(G) \geq \frac{n}{2}$ for every simple planar graph $G$ on $n$ vertices. We show that the version of this problem for multigraphs (allowing parallel edges) is easily reduced to the problem about the independence number of simple planar graphs. Specifically, we prove that $a(M) \geq \frac{n}{4}$ for every planar multigraph $M$ and that this lower bound is tight. Then, we study the case when the number of pairs of vertices with parallel edges, which we denote by $k$, is small. In particular, we prove the lower bound $a(M) \geq \frac{2}{5}n-\frac{k}{10}$ and that the Albertson-Berman conjecture for simple graphs, assuming that it holds, would imply the lower bound $a(M) \geq \frac{n-k}{2}$ for multigraphs, which would be better than the general lower bound when $k$ is small. Finally, we study the variant of the problem where the plane multigraphs are prohibited from having $2$-faces, which is the main non-trivial problem that we introduce in this article. For that variant without $2$-faces, we prove the lower bound $a(M) \geq \frac{3}{10}n+\frac{7}{30}$ and give a construction of an infinite sequence of multigraphs with $a(M)=\frac{3}{7}n+\frac{4}{7}$.
Directed Edge Geography and Undirected Edge Geography are classical PSPACE-complete two-player graph games in which players alternately make moves along edges, deleting each one after use; the first player unable to move loses. We prove that both problems are XNLP-hard when parameterized by pathwidth, addressing a question raised by Bodlaender over 30 years ago. On the positive side, we observe that Directed Edge Geography is fixed-parameter tractable when parameterized by treewidth and maximum degree. We also prove that both problems are in XP on simple graphs when parameterized by tree-partition width. These results develop modern lower-bound and decomposition-based algorithmic methods for width-based questions in PSPACE-complete graph games.
A vertex-coloring of a connected graph $G$ is a strong conflict-free vertex-connection coloring if every two distinct vertices are joined by a shortest path on which some color appears exactly once. The minimum number of colors in such a coloring is the strong conflict-free vertex-connection number $\operatorname{svcfc}(G)$. We study this problem under the parameter twin cover. Let $X$ be a twin cover of $G$ of size $t$, and let $k$ be the target number of colors. In our first result, given $(G,k)$ together with a twin cover $X$, we reduce in polynomial time to an equivalent annotated instance on at most $\max\{2,t+(t+1)k2^{t+k-1}\}$ vertices. Hence the annotated version of Strong CFVC Number, in which a twin cover is supplied as part of the input, is fixed-parameter tractable parameterized by $t+k$. Using this bound, we then obtain a kernel parameterized by $\operatorname{tc}(G)+k$; in particular, for every fixed $k$, the problem is fixed-parameter tractable parameterized by the twin-cover number alone. In our second result, we prove every connected graph $G$ with twin cover $X$ of size $t$ satisfies $χ(G)\le \operatorname{svcfc}(G)\le χ(G)+t$. More generally, if $Y\subseteq X$ intersects every shortest path of length at least $3$, then $\operatorname{svcfc}(G)\le χ(G)+|Y|$. We also derive an exact expression for the chromatic number on graphs of bounded twin-cover number: for every proper coloring $φ$ of $G[X]$, the minimum number of colors needed to extend $φ$ to all of $G$ is $K_φ=\max_{S\subseteq X}(|φ(S)|+m(S))$, and hence $χ(G)=\min_{φ\text{ proper on }G[X]} K_φ$. Our results provide the first evidence that twin cover is a useful parameter for strong conflict-free vertex-connection and show that, once a twin cover is fixed, the remaining difficulty is concentrated in a bounded additive gap above the chromatic number.
The Path Avoiding Forbidden Pairs problem (PAFP) asks whether, in a directed graph $G$ with terminals $s,t$ and a set $\mathcal{F}$ of forbidden vertex pairs, there is an $s$-$t$ path that contains at most one endpoint from each forbidden pair. We initiate the study of PAFP through a layer-based width measure. Our first focus is the union digraph $G\cup\mathcal{F}$, obtained by adding to $G$ one arc per forbidden pair, oriented according to a fixed reachability-compatible order. Let the BFS layer $L_d$ be all vertices at directed shortest-path distance $d$ from $s$, where the BFS-width from $s$ is $\max_d |L_d|$. We show if $G\cup\mathcal{F}$ has BFS-width $b$ from $s$ and only $β$ arcs going from a later BFS layer to an earlier one, then PAFP is FPT parameterized by $b+β$. The backward-arc hypothesis is essential: we show PAFP remains NP-complete when the union digraph is a DAG with BFS-width 2. We also show if the input DAG has BFS-width at most $2$ and only $k$ backward input arcs, then PAFP can be decided in $2^k |I|^{O(1)}$ time, with unrestricted forbidden pairs. This width-$2$ result is tight: inspection of a classical reduction shows NP-completeness on input DAGs of BFS-width $3$ with no backward input arcs. Moreover, we study exact-length layers in the input graph, where the $d$-th layer consists of the vertices reachable from $s$ by a directed path of length exactly $d$. For DAGs of exact-length width at most $2$, we show PAFP is polynomial-time decidable by a 2-SAT encoding of fixed-length paths. This bound is tight: the same classical reduction yields NP-completeness on DAGs of exact-length width $3$. Unlike previously known polynomial-time regimes for PAFP, which restrict the forbidden-pair set in order to obtain tractability, our two input-graph tractability results allow unrestricted forbidden pairs and input graphs with exponentially many $s$-$t$ paths.
For an $n$-vertex graph $G$, let $z(G;k)$ denote the number of zero forcing sets of size $k$. A conjecture of Boyer et al. asserts that the path $P_n$ maximizes these numbers coefficientwise among all $n$-vertex graphs; equivalently, the zero forcing polynomial of every $n$-vertex graph should be coefficientwise dominated by that of $P_n$. We prove this path-extremal conjecture for distance-hereditary graphs. This extends the previously known tree case to a much larger class that includes, in particular, all trees and all cographs. We then use canonical split decomposition to push the argument one step beyond the distance-hereditary setting. Specifically, we show that if a split-prime graph $H$ and all of its induced subgraphs are path-extremal, then every connected graph whose canonical split decomposition has a unique prime bag whose label graph is isomorphic to $H$ is also path-extremal. As a corollary, for each fixed $m$, if every induced subgraph of every split-prime graph on at most $m$ vertices is path-extremal, then so is every connected graph whose canonical split decomposition has a unique prime bag of size at most $m$. Thus, on these classes, the conjecture reduces to a finite verification problem on bounded-order prime cores. Our proofs combine two counting mechanisms for non-forcing sets -- fort obstructions arising from twin pairs and a leaf recurrence -- with the accessibility description of graph-labelled trees in the canonical split decomposition. This yields a new positive instance of the path-extremal conjecture and identifies a natural structural frontier for further progress.
In this short communication, we generalize a classical result of Barlotti concerning the unique extendability of arcs in the projective plane to higher-dimensional projective spaces. Specifically, we show that for integers \( k \ge 3 \), \( s \ge 0 \), and prime power \( q \), any \((n, k + s - 1)\)-arc in PG\((k - 1, q)\) of size \( n = (s+1)(q+1) + k - 3 \) admits a unique extension to a maximal arc, provided \( s + 2 \mid q \) and \( s < q - 2 \). This result extends the classical characterizations of maximal arcs in PG\((2,q)\) and connects naturally to the theory of A$^s$MDS codes. Our findings establish conditions under which linear codes of given dimension and Singleton defect can be uniquely extended to maximal-length projective codes.
We revisit a classic theorem of Frougny and Sakarovitch concerning automata for $φ$-representations, and show how to obtain it in a different and more computationally direct way. Using it, we can find simple, induction-free proofs of existing results in the literature about these representations, in a uniform and straightforward manner. In particular, we can easily and "automatically'' recover many of the results of recent papers of Dekking and Van Loon. We also obtain a number of new results on $φ$-representations.