Antonio Blanca, Alexandre Stauffer, Izabella Stuhl
We study the problem of sampling weighted partial triangulations of a convex polygon with $n+2$ sides. We consider the distribution $π_{n,λ}$ under which each partial triangulation $σ$ is assigned probability proportional to $λ^{|σ|}$, where $λ>0$ is a model parameter and $|σ| \in \{0,\dots,n-1\}$ denotes the number of diagonals in $σ$. This model belongs to a broad class of weighted geometric partition problems that include lattice triangulations and dyadic tilings, and is closely related to several classical combinatorial structures, including the full triangulations of a convex polygon and the associated Catalan structures. Our main result is a simple exact sampling algorithm for $π_{n,λ}$ with expected running time $O\big((\min\{n,n\sqrtλ\}+1)\log n\big)$, which is optimal up to the logarithmic factor.
The \textit{boxicity} of an undirected graph $Γ$, denoted as $\operatorname{box}(Γ)$, is the smallest non-negative integer $d$ such that $Γ$ can be represented as the intersection graph of axis-parallel rectangular boxes in $\mathbb{R}^d$. The \emph{power graph} of a group $G$, denoted as $\operatorname{Pow}(G)$, is an undirected graph whose vertex set is the group itself, with two elements being adjacent if one is a power of the other. We determine the exact boxicity of power graphs of finite nilpotent groups. For a cyclic group of order $n=\prod_{i=1}^{r}p_i^{e_i}>1$, where the $p_i$ are distinct primes, we prove that its power graph has boxicity $\sum_{i=1}^{r}e_i-\max_i e_i$. For a finite nilpotent group $G$, let the number of noncyclic odd Sylow subgroups be denoted by $t(G)$. If $G$ has odd order or if $G$ has a cyclic Sylow $2$-subgroup, then $\operatorname{box}(\operatorname{Pow}(G))=\operatorname{box}(\operatorname{Pow}(\mathbb{Z}_{\operatorname{exp}(G)}))+t(G)$, where $\operatorname{exp}(G)$ denotes the \emph{exponent} of a group $G$. When the Sylow $2$-subgroup is noncyclic, and the odd part is nontrivial, the boxicity equals this value or one more, and we characterise precisely when each of these values is attained. The boxicity of power graphs of arbitrary finite groups and the exact boxicity of the comparability graph of the Boolean lattice were posed as open questions in Discrete Mathematics, 349(7):115017, 2026. As a consequence of our results, we answer the latter question completely and the former question partially, by computing the exact boxicity of the power graphs of nilpotent groups.
A natural way to compare two graphs is to align their vertex sets by a bijection and to measure the difference of the aligned adjacency matrices in a matrix norm. Depending on the norm, this yields, for example, the edit distance, the cut distance, the spectral distance, and the local edit distance. To make these distances applicable to graphs of different order, one can consider blowups of the graphs or, equivalently, relax bijections to fractional alignments, that is, optimal transport plans between the uniform distributions on the vertex sets, in the spirit of Gromov-Wasserstein distances. We investigate how much the optimal transport versions can differ from their bijective counterparts, that is, we quantify their integrality gaps. For the edit distance, we improve the best known upper bound on the integrality gap from 3 to 2 and the lower bound from 6/5 to 4/3. For the cut distance, we approach Lovász's conjecture that $δ_\square \le 2δ_\square^{\rm OT}$ from both sides. We improve the previously best known upper bound $δ_\square \le 32 \cdot (δ_\square^{\rm OT})^{1/67}$ (Borgs, Chayes, Lovász, Sós and Vesztergombi, Adv. Math. 2008) to $δ_\square \le C_\varepsilon \cdot (δ_\square^{\rm OT})^{1-\varepsilon}$ for every $\varepsilon > 0$, and we construct graphs $G,H$ with $δ_\square(G,H) = 2δ_\square^{\rm OT}(G,H)$, establishing a factor-2 lower bound that matches Lovász's conjecture. For the local edit and spectral distances, we obtain upper bounds of order $\sqrt{t\ln(1/t)}$ and $\sqrt{t}$, respectively, where $t$ is the corresponding optimal transport distance, and we show that, unlike for the edit and cut distances, no linear upper bounds are possible. Finally, we show that computing the optimal transport distances is NP-hard, and we give hardness of approximation results for the cut, spectral, and local edit distances.
Jan Dreier, Nikolas Mählmann, Michał Pilipczuk, Szymon Toruńczyk
A graph class $\mathcal{C}$ is monadically dependent if one cannot interpret all graphs in vertex-colored graphs from $\mathcal{C}$ using a fixed formula of first-order logic. We prove that every monadically dependent graph class has almost bounded merge-width, thereby confirming a conjecture of the first and the last author [STOC '25]. Our proof is algorithmic and the witnessing decompositions in the form of merge sequences can be computed in nearly-cubic time. This is a step towards the proof of fixed-parameter tractability of model checking first-order logic on monadically dependent graph classes, which is achieved in a companion paper.
Vladimir Gurvich, Matjaž Krnc, Martin Milanič, Mikhail Vyalyi
Concepts of avoidable paths and inherent two-rooted graphs were recently introduced for the case of induced subgraphs. In this paper, we translate this framework to the setting of not necessarily induced (NNI) subgraphs, where the structural constraints, as well as conclusions, are significantly relaxed. We develop a new sufficient condition for inherence, and use it to provide complete characterizations of inherent two-rooted paths and forks. We show that for these families, the obvious necessary condition for inherence from the induced case is sufficient in the NNI setting. To complement these results, we introduce two general methods for proving non-inherence and apply them to demonstrate the non-inherence of several classes of two-rooted trees. Keywords: subgraph, two-rooted graph, two-rooted tree, avoidable two-rooted graph, inherent two-rooted graph
We study the problem of determining the maximum number of stable matchings. This classical counting problem was posed by Knuth and has been studied for five decades. We obtain a new $2.999^n$ upper bound on the number of stable matchings with $n$ men and $n$ women. This improves the best-known previous result by Palmer and Pálvölgyi. This result gets much closer to the well-known lower bound $\frac{2.28^n}{\text{poly}(n)}$. We refine their entropy-based framework by capturing broader structural relationships among stable matchings. To capture these relationships, we introduce new frameworks, including cones and supports. This establishes the $3.332^n$ bound. We also exploit relationships among local structures and use a charging argument to improve the bound to $2.999^n$.
We study the maximum number ${\rm SM}(n)$ of stable matchings in an instance with $n$ men and $n$ women. Karlin, Oveis Gharan, and Weber (STOC 2018) proved the first single-exponential upper bound; their argument gives ${\rm SM}(n)\le 2^{17n}$ for sufficiently large $n$. Palmer and Pálvölgyi (FOCS 2021) improved this bound to $3.55^n+O(1)$. We prove that ${\rm SM}(n)<2.81^n$. This brings the exponential base closer to $2.28$, the base of the best known lower bound. More precisely, we bound the number of stable matchings in each instance in terms of each person's number of stable partners, where a stable partner is someone matched to that person in at least one stable matching. Our proof assigns each person a weight determined by this number and uses these weights to bound the number of stable matchings by induction.
Amotz Bar-Noy, Omri Kosary, David Peleg, Dror Rawitz
In the Degree Realization problem with respect to a family $\mathcal{P}$ of graphs the input is a non-increasing sequence $d = (d_1,\ldots,d_n)$ of positive integers, and the goal is to decide whether there exists a simple undirected graph $G \in \mathcal{P}$, whose degrees correspond to $d$, i.e., such that $deg(G) = d$. In this paper we consider the version of Degree Realization in which the realization is required to be a forest (i.e., $\mathcal{P}$ is the family of forests). We consider \emph{optimized} Degree Realization in which the goal is to obtain a realization that minimizes an objective function $f$. That is, the goal is to find a realization $G$ that minimizes $f(G)$ among the realizations of the given input sequence. More specifically, we focus on the following functions: the size of an optimal \emph{vertex cover} and the size of an optimal \emph{dominating set}. We also consider the \emph{total} and \emph{paired} versions of both Min Vertex Cover and Min Dominating Set. We provide characterizations and linear time realization algorithms for all the above-mentioned problems. Since Min Paired Vertex Cover is closely related to Min Edge Dominating Set, our results for Degree Realization with respect to the former extend to the latter.
Let $d_1,\dots,d_m$ be directions in $\mathbb{R}^k$, where $k\ge 2$. A multiset $X$ of $n$ points of $\mathbb{R}^k$ projects to the $m$ multisets of numbers $\{\{x\cdot d_j : x\in X\}\}$, or equivalently to the $m$ sorted lists of these numbers. How many directions are needed for these projections to determine every multiset of $n$ points? Two simple bounds are known in every dimension: more than $(k-1)\log_2 n$ directions are always needed, and $(k-1)n+1$ directions in general position always suffice. In the plane much more is known: Matousek, Privetivy and Skovron showed that almost every choice of $O(\log n \log\log n)$ directions suffices. Whether anything similar holds in higher dimensions was left open by Dym, Wellershoff, Tsoukanis, Levy and Balan. We observe that two lemmas of Hair and Sahai, proved for list recovery of random linear codes, answer that question, and by tightening their argument we show that in every dimension almost every choice of $(4k-2)\ln(2n)\ln(k\log_2 2n)+k+1$ directions suffices. For $k\le 10$ this beats the linear bound once $n$ is larger than about 100: for example, 144 directions suffice for multisets of 1000 points in the plane, and an exact computation lowers this to 133. Letting some of the random blocks in their argument fail removes a factor $\log k$: almost every choice of $O(k\log n\log\log n)$ directions suffices, uniformly in $k$ and $n$. We also show that $2k-1$ directions never determine three points, so that the logarithmic lower bound is not tight; that every $m$ directions in the plane fail on some pair of multisets of size $O(1.78^m)$, improving the bound $O(1.80^m)$ of Matousek, Privetivy and Skovron; and that for each $k$, $m$ and $n$, either almost every choice of $m$ directions works or almost none does.
Esther Galby, Roohani Sharma, Bart van der Steenhoven
In [Combinatorica 1996], Reed, Robertson, Seymour and Thomas proved Younger's conjecture dating back to 1973 that states that directed cycles have the Erdős-Pósa property. Specifically, they showed that every directed graph $D$ contains either $k$ arc-disjoint cycles or a set of at most $f(k)$ arcs that intersect every directed cycle. The function $f$, depending only on $k$, is however notoriously large--in fact, not even elementary. The first improvement on this function was only very recently provided by Hatzel, Kreutzer, Milani, and Muzi [FOCS 2024] who showed that $f$ can be taken to be an exponential tower of height $8$. In this work, we improve this function to $O(k^4)$ in the case where the digraph is Eulerian. This also answers an open question by Wahlström [Springer Monographs in Mathematics 2018] who asked explicitly when Eulerian digraphs have a polynomial Erdős-Pósa property. The only other known result about Erdős-Pósa on Eulerian digraphs is a result from Seymour [Combinatorica 1996] who showed that $f(k)=k$ when the Eulerian digraph admits a linkless embedding in 3-space, and such an equality is not possible for general Eulerian digraphs. We also complement our positive result, by showing that the result of Seymour cannot be extended to give a linear bound ($O(k)$) on general Eulerian digraphs, by showing that even on Eulerian digraphs the lower bound for the Erdős-Pósa property is $Ω(k \log k)$.
Every fixed first-order sentence $\varphi$ determines an enumerative sequence $n\mapsto \mathrm{FOMC}(\varphi,n)$, counting its models on the labeled domain $[n]$. We study the complexity of these sequences when logical specifications may use genuine unary function symbols and hence nested terms such as $x,f(x),f^2(x),\ldots$. We prove that, for every fixed sentence $\varphi\in \mathrm{C}^1_{=}[f]$ with one unary function and an arbitrary finite relational vocabulary, $\mathrm{FOMC}(\varphi,n)$ is computable in time polynomial in $n$. By contrast, either a second variable or a second unary function yields hardness: there is a fixed sentence in $\mathrm{FO}^2_{=}[f]$ whose model-counting function is $\#\mathrm{P}_1$-complete, and there is a fixed constant-free universal sentence in $\mathrm{FO}^1_{=}[f,g]$, using only unary predicates besides $f$ and $g$, with the same property. We also relate labeled and unlabeled enumeration exactly. For every relational sentence $\varphi$, we construct an extension $\varphi_{\mathrm{aut}}$ in which a unary function records an automorphism and $\mathrm{FOMC}(\varphi_{\mathrm{aut}},n)=n! \cdot \mathrm{UFOMC}(\varphi,n)$, where $\mathrm{UFOMC}(\varphi,n)$ counts the $n$-element models of $\varphi$ up to isomorphism. This gives a one-query exact reduction from unlabeled to labeled model counting at the same domain size. If $\varphi$ belongs to $\mathrm{FO}^k_{=}$, respectively $\mathrm{C}^k$, and its relational vocabulary has maximum arity $r$, eliminating the auxiliary function yields a single-query reduction to labeled model counting in $\mathrm{FO}^{\max\{k,r+1\}}_{=}$, respectively $\mathrm{C}^{\max\{k,r+1\}}$.
We view a $k\times n$ Latin rectangle as $n$ points in the grid $[n]^k$. Each symbol gives one point, whose coordinates are the columns it occupies. The rectangle is balanced when these points are pairwise equidistant in the $\ell_1$ metric. Its imbalance measures the distance from this ideal. We study the minimum imbalance from three sides. Constructively, we use a simulated annealing that moves symbols within rows and only penalises column clashes. Against the reference table of Voboril et al. (2025), it improves 33 of the 51 non-square values for $8\le n\le 13$, by up to 39.4%, and matches the rest. A cheap invariant, the row group, certifies that none of the 33 new rectangles is isotopic to rows of a group table, so the earlier column-swapping local search cannot reach any of them. Exactly, a computer-assisted branch and bound proves that the minimum imbalance of a $3\times 9$ rectangle is $56$. Díaz et al. found this value, but its optimality was open. A sharper pruning bound, derived from the identity $I=2\sum(d-t)^+$, reduces the search tree by a factor of $2.4$ to $3.3$ on the instances with $k\ge3$ that we solve. Asymptotically, the minimum imbalance is $Θ(n^3)$ for every fixed $k$. A pigeonhole argument gives a lower constant of order $k\,4^{-k}$, and random rectangles give an upper constant of order $\sqrt k$. For $k=2$, the limit of $\min I/n^3$ exists, a permuton attains it, and it lies between $4/375$ and $0.1196$. Perfect balance is an $\ell_1$-equilateral problem, so it forces $k=Ω(n/\log n)$ rows. For $2\times n$ rectangles, single two-minute annealing runs improve five of the seven six-hour MIP upper bounds for $14\le n\le 20$. Finally, the Lean 4 kernel checks every rectangle reported in this paper against a 74-line specification that imports only Lean's core library. We also prove the pigeonhole lower bound in Lean with Mathlib.
We resolve longstanding questions about connectivity, reconstruction,and looseness in random $k$-SAT, for sufficiently large fixed $k$ as the number $n$ of variables tends to infinity. $\textbf{Single-variable connectivity up to reconstruction.}$ At clause densities up to \[\frac{2^k}{k}\bigl(\ln k+\ln\ln k+γ_*-o_k(1)\bigr),\] a polynomial-time algorithm joins a typical pair of satisfying assignments with high probability, using $O_k(n)$ single-variable flips that preserve satisfiability. Here $γ_*\approx0.871$ arises from a refined tree-reconstruction analysis. $\textbf{Settling the reconstruction threshold.}$ Building on the tools developed for connectivity, we determine the clustering threshold defined by the reconstruction criterion to be \[\frac{2^k}{k}\bigl(\ln k+\ln\ln k+γ_*+o_k(1)\bigr),\] below the classical cavity prediction with constant term $1+O(\ln\ln k/\ln k)$. $\textbf{Shattering before rigidity.}$ We locate the looseness--rigidity transition at a higher density, matching the classical cavity prediction for clustering: \[\frac{2^k}{k} \left(\ln k+\ln\ln k+1+Θ\left(\frac{\ln\ln k}{\ln k}\right)\right).\] Below the transition, we establish a stronger form of looseness: any variable of a typical solution can be reversed by logarithmically many single-variable flips that preserve satisfiability. Above it, reversing most variables of a typical solution requires changing linearly many variables (rigidity). Yet throughout an interval between reconstruction and rigidity, the solution space already splits into groups of exponentially small relative size separated by linear Hamming distance and linear energy barriers (shattering), while logarithmic looseness persists.
We study metric distortion in one-sided matching. Given $n$ agents and $n$ items in an unknown metric space, our goal is to match agents to items so that the total distance between matched pairs is small. However, instead of direct access to the distances in the metric, we are only given each agent's ranking of the items by distance. What is the smallest worst-case approximation factor (called the distortion) that a mechanism can guarantee in expectation? Despite significant interest over the last decade, the best-known guarantee has been the $O(n)$ distortion of Random Serial Dictatorship since the beginning. We give a matching mechanism with distortion $O(n^{\log_4 3})$, the first sublinear upper bound. The key idea is to use a partial matching derived from a truncated version of either Random Serial Dictatorship or the Probabilistic Serial rule. We then complete this matching using a new, general-purpose cluster completion algorithm that ensures unmatched agents receive nearby items. Along the way, we pin down the optimal distortion of Random Serial Dictatorship and Probabilistic Serial up to logarithmic and constant factors, respectively, and give improved lower bounds for truthful mechanisms.
The classification of K-geodetic graphs is a fundamental problem in graph theory concerning the number of shortest paths connecting pairs of vertices. A graph is called K-geodetic if every pair of distinct vertices is joined by at most K geodesics, where a geodesic is a shortest path between two vertices. The cases K=1,2,3 correspond, respectively, to geodetic, bigeodetic, and trigeodetic graphs. Despite substantial progress in the study of their structural properties, a comprehensive classification of these graph families remains elusive. This paper develops a research framework for investigating the classification problems by combining metric graph theory, distance-regular graphs, algebraic graph theory, and combinatorial design theory. Particular attention is given to distance-regular graphs, for which the number of geodesics between vertices at a fixed distance is determined by the intersection numbers of the graph. This property converts the K-geodetic condition into a finite collection of arithmetic inequalities. We establish the resulting necessary conditions for K=1,2,3, discuss representative examples, and distinguish these conditions from the stronger requirements needed to prove the existence and classification of graphs with a given intersection array. The paper also investigates the potential role of balanced incomplete block designs (BIBDs) in constructing and organizing families of K-geodetic graphs. Finally, it proposes a computational research methodology based on constraint satisfaction, graph-isomorphism testing, exhaustive enumeration, and machine-assisted conjecture generation.
Many combinatorial designs ask for equal distribution of given symbols across the entries of a matrix. The paramount examples are Latin squares, where each symbol from $\{1,\dots,n\}$ appears once per row and column of an $n\times n$ matrix. Frequency rectangles extend this to $λn \times μn$ matrices with repeated symbols under controlled column and row frequencies. In this more general setting, we examine structural properties of pairwise reflection-symmetry, which requires that, on every pair of columns, each ordered symbol pair $(p,q)$ occurs as often as its reversal $(q,p)$. This order-balance is precisely what makes head-to-head comparisons unbiased, i.e., no symbol gains a systematic advantage from the position it occupies relative to another, a fairness demand arising for instance when scheduling tournaments or laying out comparative trials. Existence of such objects for odd $λ$ turns out to be remarkably more subtle than for even $λ$. After showing that existence holds also for sufficiently large odd $λ$, we initiate the search for the smallest possible value of $λ$ in this setting. We obtain the insight that a per-column multiplicity of $λ=1$ can be achieved if and only if $n$ is a power of two. We complement the existence results with a direct product construction and add several further observations on the property. Finally, we propose and evaluate a quadratically constrained integer program to computationally search for these objects. The resulting experiments reveal that many of them possess an underlying group-theoretic structure which, as we conjecture, may even be unavoidable in certain settings.
The chromatic index $χ'(H)$ of a hypergraph $H$ is the smallest number of colors needed to color its edges so that no two adjacent edges share the same color. The chromatic index is trivially bounded below by the maximum degree $Δ(H)$, and when $χ'(H)= Δ(H)$, we say that $H$ is Class 1. It is well known that almost every 2-graph is Class 1, and we extend this result to 3-graphs. We show for $n^{-2/3-1/200}\leq p \leq 0.99$ that the $3$-uniform binomial random hypergraph $H\sim H_{n,p}^{(3)}$ is Class 1 with high probability. We also show that this result doesn't hold for uniformities above $3$. For a broad range of $p$, when $k \geq 4$ and $k \nmid n$, the hypergraph $H \sim H_{n,p}^{(k)}$ is with high probability not Class 1.
Transfer of Weisfeiler-Leman equivalence under graph operations
Dmitriy Kunisky, Hamilton Sawczuk
AI 速览：研究图运算Φ下WL等价关系的传递参数τ_Φ(k)，并精确确定线图及(r,S)-线图的传递参数。
The Weisfeiler-Leman algorithms WL(k) for $k\ge1$ form a hierarchy of increasingly powerful tests for graph isomorphism. We study how the equivalence relations they induce on graphs transfer through a graph operation $Φ$: for a given $k$, what level of WL-equivalence between graphs $H_1$ and $H_2$ guarantees WL(k)-equivalence of graphs $Φ(H_1)$ and $Φ(H_2)$? We call the least such level the transfer parameter $τ_Φ(k)$ and determine it exactly for the line graph operation $Φ= L$: $τ_L(k)=2k+1$ for $k\ge2$, while $τ_L(1)=1$. We also extend these results to the $(r,S)$-line graphs, which generalize the line graph from edges to $r$-cliques by connecting two cliques when their intersection size lies in a prescribed set $S$. For these operations we prove that $τ_{r,S}(1)=r-1$ for every $S$ and, for certain choices of $S$, that $τ_{r,S}(k)=rk+r-1$ for $k\ge r$, and we bound the transfer parameter in the remaining range. Our upper bounds on transfer parameters follow from a treewidth analysis of diagrammatic expansions of homomorphism counts into $Φ(H)$ as linear combinations of homomorphism counts into $H$. The matching lower bounds combine explicit constructions of witness graphs with the homomorphism-distinguishing closedness of graphs of bounded treewidth.
On the Power of Determinism in Multi-Item Auctions
Yiannis Giannakopoulos, Johannes Hahn
AI 速览：作者研究单一加性买家、多件独立非同分布物品的垄断拍卖，通过拉格朗日对偶上界改进三种简单确定性拍卖相对最优真实拍卖收益的近似比。
We study the classical multi-item monopoly setting with a single additive buyer and $m$ heterogeneous items whose values are independent but not necessarily identically distributed. We compare the revenue (REV) of the optimal truthful auction, which in general may be randomized or complicated, with the revenues of three simple deterministic auctions: selling all items separately (SREV), selling them as a single grand bundle (BREV), and choosing the better of the two. Our technical cornerstone is a general upper bound on the optimal multidimensional revenue, parameterized by freely chosen item-wise caps, and expressed in terms of optimal single-item, Myersonian revenues and basic statistics of the value distributions. We derive this inequality through Lagrangian duality for a nonlinear programming formulation of the worst-case approximation ratio of separate-selling. Combining this inequality with carefully chosen caps and a probabilistic analysis of the resulting terms, we obtain improved approximation guarantees for all three deterministic auctions, both for general independent values and for iid values. Notably, we prove that $REV\le 2.5\max\{SREV,BREV\}$, improving the long-standing $5.2$ factor of Ma and Simchi-Levi and the $6$ factor of Babaioff, Immorlica, Lucier, and Weinberg. We complement this upper bound with a construction establishing a lower bound of $2.19$, improving the bound of $2$ by Rubinstein. For two iid items, we determine the exact approximation ratio of selling separately. Our proof constructs tight Lagrangian dual certificates for the aforementioned programming formulation, when item values lie in the grid $\{0,1/K,2/K,\dots,1\}$, for any discretization parameter $K$. Taking $K\to\infty$ yields the tight ratio $1+W(1/e)\approx1.2785$ in the continuous-valued setting, where $W$ is the Lambert-W function, closing the $[1.278,1.368]$ gap left by Hart and Nisan.
Prosenjit Bose, Vida Dujmović, Hussein Houdrouge, Pat Morin, Saeed Odak
A dominating set of a graph $G$ is \emph{connected} if it induces a connected graph in $G$. For planar triangulations, it has been known since 1990 that every $n$-vertex triangulation with $n\ge4$ admits a connected dominating set of size at most $n/2 - 1$, and no improvement to this bound was known for over three decades. We break this longstanding barrier by showing that every $n$-vertex triangulation has a connected dominating set of size at most $10n/21$. Equivalently, every triangulation admits a spanning tree with at least $11n/21$ leaves. This result has graph drawing applications, establishing a bound for one-bend free sets and improving the known bound for simultaneous planar embeddings. The connected dominating set results extend readily to show that an $n$-vertex triangulation of a surface of Euler genus $g$ has a connected dominating set of size at most $10n/21+O(\sqrt{gn})$.
Using the bisection bound of Monien and Preis, Fomin and Høie proved in 2006 that every graph of maximum degree 3 on $n$ vertices has pathwidth at most $n/6+o(n)$. We prove that the pathwidth, bisection width and cutwidth of these graphs are at most $c_3 n+o(n)$, where $c_3=\frac{3}{4π}\arccos\frac56\approx0.1398$. For bisection width this improves the bound $n/6+o(n)$ of Monien and Preis (2006) and extends to all such graphs the bound that Lichev and Mitsche (2023) obtained for random cubic graphs. For maximum degree 4, we prove that bisection width and cutwidth are at most $n/3+o(n)$, improving the bound $2n/5+o(n)$ of Monien and Preis for 4-regular graphs, and that pathwidth is at most $0.2458n$, improving the bound $n/3$ of Fomin, Gaspers, Saurabh and Stepanov (2009). We show that every graph with $m$ edges has pathwidth at most $7m/48+o(n)$, improving the bound $13m/75$ of Scott and Sorkin (2007), and we give further bounds of the form $\sum_{d\ge3}a_d n_d+o(n)$ with $a_d<1$, where $n_d$ is the number of vertices of degree $d$. The path decompositions, and linear orderings of the stated cutwidth, can be computed in polynomial time. Instead of deriving pathwidth bounds from bisection width, we give each vertex a Gaussian label, a weighted sum of independent standard normal variables over a ball around it, and sweep a threshold through the labels; the bags are frontiers, the vertices above the threshold with a neighbor below it. We bound the probability that a vertex lies in a frontier by comparison with an explicit Gaussian model of a vertex and its neighbors. Faster exact algorithms follow: for feedback vertex set and the traveling salesperson problem on graphs of bounded maximum or average degree, and on general graphs for minimum weight dominating set (by branching until the instance is sparse), Max 2-CSP and counting subgraphs with $k$ edges (by the bound in $m$).
Ralf Klasing, George B. Mertzios, Hendrik Molter, Nils Morawietz, Paul G. Spirakis
In classical gossiping, each participant initially knows a distinct piece of information. Whenever two participants call each other, they exchange everything they know. We study Gossiping: given an undirected multigraph whose vertices represent participants and whose edges represent available calls, decide whether the calls can be ordered so that every participant learns all information, using each edge at most once. Equivalently, can one time label per edge make the graph temporally connected by strict temporal paths? This recognition problem is NP-complete; we study its parameterized complexity. After linear-time preprocessing, we give an exact $O(3^{n/3}n^3)$-time algorithm and a deterministic $2^{O(tw)}n$-time algorithm, where $n$ is the number of vertices and $tw$ is the treewidth of the underlying simple graph. We introduce a Tree Contraction Rule that extends the treewidth algorithm to single-exponential algorithms for parameters that may remain small on dense graphs, such as tree-independence number and clique-width, without requiring a decomposition as input. For the excess $δ=m-(2n-4)$ over the minimum number of calls, where $m$ is the number of edges, we give a randomized $2^{O(δ)}n^3$-time algorithm with one-sided error. Its main subroutine solves Rooted Independent Cycle: finding a cycle through four prescribed vertices whose edges are independent in a represented linear matroid. We solve this problem deterministically in $2^{O(r)}m$ arithmetic operations over the finite field representing the matroid, where $r$ is its rank. Gossiping is polynomial-time solvable at maximum degree four, counting parallel edges, but NP-complete on planar simple graphs of maximum degree five. The treewidth algorithm yields a $2^{O(\sqrt n)}$-time algorithm on planar graphs.
In 2010 Banderier and Nicodeme consider the height of bounded discrete bridges and conclude to a limiting Rayleigh distribution. This result is correct although their proof is partly erroneous. They make asymptotic simplifications based upon dominance properties of the roots of the kernel of the walk within a disk centered at the origin, but these dominance properties apply only upon a positive real segment. However the very good agreement of simulations with their asymptotic expansion of the probability distribution in case of Łukasiewicz bridges let us think that their proof could be corrected. This is the scope of the present article which provides a proof using the dominance property only in its domain of validity. We also consider the case of periodic walks, a topic not considered in Banderier-Nicodeme2010..
The Nash-Williams--Tutte theorem characterizes the existence of $k$ edge-disjoint spanning trees by a condition on every vertex partition. We prove that, for simple graphs, the same condition guarantees $k$ edge-disjoint spanning trees, each of diameter $O(n/δ)$ and with $O((n/δ)\log(δ+1))$ internal vertices, where $n$ is the number of vertices and $δ$ is the minimum degree. Both bounds hold simultaneously for every tree, with absolute implicit constants independent of $k$, and each is asymptotically optimal in general. Our proof is constructive and yields a randomized algorithm with expected running time $\tilde{O}(n^5)$. We derive applications to short edge-disjoint routes and weak linkages, small Steiner trees, and survivable networks with short paths. The joint bounds also yield low-diameter spanning Eulerian subgraphs and short tours for postman and general routing problems.
A proper circular-arc (PCA) model is a pair $M = (C, A)$ where $C$ is a circle and $A$ is a family of inclusion-free arcs on $C$ whose extremes are pairwise different. The model $M$ represents a digraph $D$ that has one vertex $v(a)$ for each $a \in A$ and one edge $v(a) \to v(b)$ for each pair of arcs $a,b \in A(M)$ such that the beginning point of $b$ belongs to $a$. For $k \geq 0$, the $k$-th power $D^k$ of $D$ has the same vertices as $D$ and $v(a) \to v(b)$ is an edge of $D^k$ when $a\neq b$ and the distance from $v(a)$ to $v(b)$ in $D$ is at most $k$. A unit circular-arc (UCA) model is a PCA model $U = (C,A)$ in which all the arcs have the same length $\ell+1$. If $\ell$, the length $c$ of $C$, and the extremes of the arcs of $A$ are integer, then $U$ is a $(c,\ell)$-CA model. For $i \geq 0$, the model $i \times U$ of $U$ is obtained by replacing each arc $(s,s+\ell+1)$ with the arc $(s,s+i\ell+1)$. If $U$ represents a digraph $D$, then $U$ is $k$-multiplicative when $i \times U$ represents $D^i$ for every $0 \leq i \leq k$. In this article we design a linear time algorithm to decide if a PCA model $M$ is equivalent to a $k$-multiplicative UCA model when $k$ is given as input. The algorithm either outputs a $k$-multiplicative UCA model $U$ equivalent to $M$ or a negative certificate that can be authenticated in linear time. Our main technical tool is a new characterization of those PCA models that are equivalent to $k$-multiplicative UCA models. For $k=1$, this characterization yields a new algorithm for the classical representation problem that is simpler than the previously known algorithms.
Marco Bressan, Stefano Clemente, Giacomo Fumagalli, Daniel Paul-Pena, C. Seshadhri
A hypergraph $H = (V,E)$ describes a set system over a vertex universe $V$. Hypergraphs are increasingly popular in network science, as they represent interactions between subsets of vertices. A natural challenge is to design a data structure for $H$ to quickly answer basic questions about it, e.g. how many neighbors a vertex $v$ has or whether a hyperedge contains a vertex subset $S$. Such questions are trivial for graphs, for which we simply inspect the adjacency list. The gold standard, from a data structures perspective, would be near-linear time preprocessing and query answering in (poly)logarithmic time. Since this is unlikely for arbitrary $H$, we ask for well-motivated hypergraph properties that can make such a goal possible. We introduce bidegeneracy, a measure of hypergraph sparsity that strictly generalizes many other measures, including bounded degree, bounded rank, and bounded $\infty$-degeneracy. We show that bounded bidegeneracy yields fast hypergraph data structures. For a large set of fundamental queries, including their listing and sampling versions, we obtain near-linear preprocessing and (poly)logarithmic query time. Moreover, direct reductions from assumptions like the Orthogonal Vectors Conjecture suggest that, without assumptions on the input, these structures require almost-quadratic preprocessing or almost-linear query time. We use bidegeneracy to compute a compressed representation of $H$ and perform careful reductions to stabbing and searching problems in $\mathbb{R}^d$. Our framework significantly extends the well-studied "partial matching" problem of finding database entries through wildcard search. It also uncovers an intriguing connection between hypergraph sparsity, compression, and the complexity of hypergraph primitives. We also find remarkably low bidegeneracy in hypergraph datasets from the literature, suggesting practical consequences.
We give a conic upper bound on the asymptotic rank of every weighted graph tensor over every field. If $W$ is the logarithmic edge-weight matrix of its dimension matrix $Q$, then $$ \operatorname{MaxCut}(W)\le\log_2\operatorname{\widetilde R}_{\mathbb F}(\operatorname{WT}(Q))\le U_{\operatorname{CP}}^Δ(W). $$ Here $U_{\operatorname{CP}}^Δ(W)$ maximizes $\sum_{1\le i<j\le n}W_{i,j}P_{i,j}$ over nonnegative symmetric zero-diagonal matrices $P$ such that $J-2P$ is copositive and $P_{i,j}\le P_{i,k}+P_{k,j}$ for all distinct vertices. The matrix $J$ is all-one, and copositivity means nonnegativity of the quadratic form on nonnegative vectors. Our starting point extends OpenAI's polynomial multiplication method and its subsequent arbitrary-field developments from three tensor legs to arbitrary order. We prove $$ \left\lfloor\frac{n^2}{4}\right\rfloor\le\log_2\operatorname{\widetilde R}_{\mathbb F}(\operatorname{GT}_2(K_n))\le\frac{n^2}{4}. $$ The edge exponent is at most $n/(2(n-1))$ and tends to the optimal value $1/2$, improving the previous Coppersmith--Winograd bounds and answering Question 1.3.1 of Christandl, Vrana, and Zuiddam. The bounds coincide for every even $n$. For every odd $m\ge3$, the theorem gives $\log_2\widetilde R_{\mathbb F}(\operatorname{GT}_2(C_m))\le m-1+1/(m+1)$, improving the previous cycle bound with a gap of only $1/(m+1)$ above the maximum cut. On weighted triangles, the program recovers the rectangular matrix multiplication curve of Uznański and Amanov, van den Berg, and Zuiddam. The CP and semidefinite programs, with or without path constraints, all have value $9/4$ on the unweighted triangle, giving this upper bound a natural conic interpretation. Further applications give explicit bounds for complete multipartite graphs, split graphs, and wheels, including new cases of maximum-cut equality.
An orientation of a graph is called Pfaffian if all perfect matchings occur with the same sign in its Pfaffian expansion. A graph is Pfaffian if it admits such an orientation. Given a Pfaffian orientation, the number of perfect matchings can be computed in polynomial time. In 1963, Kasteleyn showed that every planar graph is Pfaffian. In 1989, Vazirani and Yannakakis showed that deciding whether a graph is Pfaffian is polynomial time equivalent to testing whether a given orientation is Pfaffian. For bipartite graphs, polynomial time recognition algorithms were obtained by Robertson, Seymour, and Thomas in 1999, and independently by McCuaig in 2004. Whether a deterministic polynomial time algorithm exists for general graphs remains open. In this paper, we first construct a polynomial that is identically zero if and only if a given orientation is Pfaffian, and show that this polynomial can be evaluated in polynomial time. Combined with the reduction of Vazirani and Yannakakis, this gives a randomized polynomial time algorithm for deciding whether a general graph is Pfaffian, placing the problem in $\mathsf{coRP}$. We also prove that an orientation of a matching covered graph is non-Pfaffian if and only if reversing the edges of some cut makes it strongly connected. Although deciding whether such a cut exists is NP-complete for general digraphs, our characterization yields a randomized polynomial time algorithm for matching covered graphs and deterministic polynomial time algorithms for bipartite and planar matching covered graphs.
We show that Max Independent Set (MIS) remains NP-hard in graphs excluding the $5 \times 5$ grid as an induced minor. This negatively answers a question of Dallard--Milanič--Štorgel that has attracted sustained attention, and refutes a conjecture by Gartland and Lokshtanov. Unless $N$-variable 3-SAT can be solved in time $2^{\mathcal O(N^{1-\varepsilon})}$ for some $\varepsilon > 0$, MIS requires $2^{Ω(n^{1/2-o(1)})}$ time on $n$-vertex graphs in this class. This complements a $2^{\widetilde{O}(n^{2/3})}$-time algorithm on any $H$-induced-minor-free class by Korhonen and Lokshtanov. Exploiting a connection between low twin-width or reduced parameters on one hand and planar-induced-minor-freeness on the other, we further show that 3-Coloring, Feedback Vertex Set and Dominating Set are NP-complete on weakly sparse graphs of reduced component max-leaf (hence twin-width) at most 3, even when a witnessing contraction sequence is given. Classes of bounded reduced component max-leaf (by any constant) and of twin-width at most 3 are known to exclude a fixed planar graph as an induced minor. The 3-Coloring and Feedback Vertex Set results refute more cases of the Gartland--Lokshtanov conjecture, even when the inputs are weakly sparse. The complexity of MIS in weakly sparse graphs excluding a planar induced minor remains open.
We show that for every $γ>0$ and every sufficiently large $Δ$, the Glauber dynamics for proper $q$-colorings of any $n$-vertex graph of maximum degree $Δ$ has spectral gap $Ω_γ(1/n)$, and hence mixing time $O_γ(n^2\log q)$, whenever $q\ge(1+1/\mathrm{e}+γ)Δ$. Our proof establishes a Bochner-type inequality by exploiting the spectral gaps of hard-core models hidden in the color classes. The constant $1+1/\mathrm{e}\approx1.368$ comes from the uniqueness threshold $\approx\mathrm{e}/Δ$ of the hard-core model on the infinite $Δ$-regular tree.