Papers

Let $D$ be a finite simple minimally strongly $k$-connected digraph, where $k\ge 2$, and let $n=|V(D)|$. Mader proved in 1985 that $|A(D)|\le 2k(n-k)$ for all sufficiently large $n$, with equality if and only if $D\cong \overleftrightarrow{K}_{k,n-k}$ (JCTB 1985). We prove that this bound holds whenever $n\ge 3k-1$ and that the order threshold $3k-1$ is best possible. Moreover, if $n\ge 3k$, then equality holds if and only if $D\cong \overleftrightarrow{K}_{k,n-k}$. The threshold $3k$ is also best possible. These results resolve the conjecture of Mader in (JCTB 1985).

We study a non-convex, nonlinear integer mathematical program with a linear constraint. This problem originates in finance and consists of selling a large predetermined quantity of a financial asset over a fixed time horizon, while maximizing the total proceeds. The selling price is modeled using a deterministic penalty function capturing the market's adverse reaction to the selling flow. From a complexity perspective, the problem is in NP. We first solve the problem exactly using dynamic programming and propose several heuristics that yield tight numerical lower bounds. We then reformulate the problem into an equivalent factorable form and derive a tight convex relaxation using McCormick inequalities. This relaxation is embedded within an adapted Branch & Bound algorithm to find the global optimum in finite time. Numerical experiments show that our methods outperform several widely used generic solvers (LocalSolver, Scip, Baron, Couenne, and Bonmin) for selected instances, but do not yield tight upper bounds, beyond small instances.

Let $G$ be a finite bipartite graph and let $\widetilde G$ be an $M$-fold cover. We compare independent sets of $\widetilde G$ with those of the trivial cover, consisting of $M$ disjoint copies of $G$, after prescribing how many vertices are selected above each vertex of $G$. A conjecture from 2011, motivated by the Bethe approximation, asserts that the trivial cover maximizes every such count. We prove this for every feasible occupation profile whose active graph has treewidth at most two or is chordal bipartite. Equality holds exactly when the restricted cover is flat, meaning that fibre relabelings make it trivial. For $M\ge2$ and positive feasible occupations, a nonflat cover of a connected bipartite graph of treewidth at most two has a deficit of at least twice the coefficient of $p_G^{M-2}$ with every occupation reduced by one. A holonomy-dependent bound improves the factor to three for odd-order holonomy; both uniform constants are sharp. Comparison and rigidity survive finite compositions along an edge or a vertex, and quantitative bounds from one piece propagate to the whole graph, giving results beyond the two original classes. On any bipartite base, the first nonzero homogeneous defect is determined by the shortest cycles with nontrivial holonomy. At singleton occupations, our counts are numbers of DP-colorings of full covers. The dual DP color function on these classes and their finite seam compositions equals the chromatic polynomial, with equality attained precisely by canonical labelings. Dong's chromatic-polynomial inequality then yields the Dual-DP Shameful inequality on the same graph class. The main theorems have been formalized in Lean 4.

In this paper we study pairwise compatibility graphs whose witness tree is required to be an unweighted caterpillar. This is a natural restriction of PCGs, in the same line of research as the study of star witnesses, which led to the class of double-threshold graphs. In the caterpillar case, we show that this restriction is equivalent to a one-dimensional difference-labelling problem: a graph admits such a representation if and only if its vertices can be assigned integer labels so that adjacency is determined exactly by whether the absolute difference of two labels belongs to a given interval. Using this characterization, we derive structural properties of the class, including bounded twin-width, and study its relation with familiar graph families. We show that the class contains cycles and two-dimensional grids, while it excludes natural obstructions such as the subdivided star $S_{2,2,2,2,2}$, the cube graph $Q_3$, the wheel $W_6$, and the complement of the $3$-sunlet. We also consider restricted representations, including centipede witnesses and representations with a given interval and we give a polynomial-time recognition algorithm when the interval is fixed.

A digraph $D = (V,E)$ is 3-transitive (respectively, 3-quasi-transitive) if whenever $(u,v,w,z)$ is a directed path of length 3, $(u,z) \in E$ (respectively, $(u,z) \in E$ or $(z,u) \in E$). These digraphs extend the concepts of transitive and quasi-transitive digraphs, respectively. For 3-transitive and 3-quasi-transitive digraphs, only the strong case has been characterized. In this paper, we introduce the notion of semi-acyclic digraphs and use it to describe completely the structure of 3-transitive and 3-quasi-transitive digraphs. As an application of this type of structure description, we prove that the path partition conjecture holds for the class of extended semi-acyclic digraphs, which generalizes two previously known results on the conjecture.

Brambles of constant congestion in directed graphs have interesting structural and algorithmic applications, and it is desirable to find the smallest function $f_c(k)$ such that if a digraph has directed treewidth at least $f_c(k)$, then it contains a bramble of size at least $k$ and congestion at most $c$. By the Directed Grid Theorem of Kawarabayashi and Kreutzer [STOC 2015], such a function $f_c(k)$ exists for every value $c \geq 2$ of the congestion. However, even using the recent improvements due to Hatzel et al. [FOCS 2024], the resulting function is a power tower of height $22$. Recently, Masařík et al. [SIAM J. Discrete Math. 2022] proved that, for congestion $8$, the function $f_8(k)$ can be made polynomial; more precisely, they showed that $f_8(k) = \mathcal{O}(k^{48} \cdot \log^{13} k)$. They asked whether it is possible to decrease the congestion, ideally to $2$, even at the cost of a higher polynomial function. In this work we answer their question affirmatively with the best possible value of the congestion, by proving that $f_2(k) = \mathcal{O}(k^{96}\cdot \log^{23} k)$. We also obtain smaller polynomials for higher congestion, namely $f_4(k) = \mathcal{O}(k^{72} \cdot \log^{21} k)$ and $f_6(k) = \mathcal{O}(k^{66} \cdot \log^{18} k)$. Our results are inspired by the proof for congestion $8$, which we improve by introducing walkages. Walkages allow us to avoid technicalities of the original proof by translating ideas due to Reed and Wood [Eur. J. Comb. 2012] that work for undirected graphs to the directed setting. Our proofs are constructive and naturally yield polynomial-time algorithms to find the brambles assuming a certificate for large directed treewidth is given. Moreover, we discuss algorithmic applications of our results for the Directed Disjoint Paths with Congestion problem, and to half-integral packings of subgraphs with large directed treewidth.

Equitable subpartitionability (ESP) is a sufficient condition for an integral system to be box-totally dual integral. We show that the converse fails even when alternative descriptions of the same polyhedron are allowed. We provide a geometrical characterization of polyhedra that can be described by ESP systems in terms of normal cones. We construct a four-dimensional integer polytope described by a totally equimodular matrix that admits no integral ESP description. This answers negatively a previously posed question by the same authors.

The small blocking set detection and construction problem asks whether a given connected simple graph contains a nonempty independent set whose cardinality equals the size of its neighborhood and is strictly less than half the number of vertices, and, if so, how to construct such a set. Previous work characterized restless explorability in vertex-permuted temporal graphs using small blocking sets, but left the computational complexity of detecting their existence as an open problem. We give a polynomial-time algorithm for detection and construction. Specifically, we first compute a maximum-cardinality critical independent set and use its difference and cardinality to handle the cases that admit an immediate solution. In the remaining boundary case, where the critical difference is zero and the selected independent set contains exactly half the vertices, we establish a projection theorem using a perfect matching and neighborhood counting: the graph contains a small blocking set if and only if the selected independent set contains a nonempty proper tight subset. We then use matching contraction to translate the tight-subset condition into the absence of arcs leaving a vertex subset in an auxiliary digraph. This yields a strong-connectivity test for the existence of a small blocking set and, when one exists, a construction based on a sink strongly connected component. The result resolves the open complexity question for small blocking set detection and provides an explicit construction algorithm.

A geometric $(n_4)$ configuration is a set of $n$ points and $n$ lines in the real projective plane such that every point lies on exactly four of the lines and every line contains exactly four of the points. The question of which $n$ admit such a configuration was posed by Grünbaum in 2000, following his 1990 paper with Rigby that gave the first geometric $4$-configuration, a $(21_4)$. Successive work of Bokowski-Schewe, of Bokowski, Grünbaum and Schewe, of Bokowski-Pilaud, and of Cuntz settled every case but one. Cuntz's 2018 paper, which constructed $(22_4)$ and $(26_4)$ configurations, states that only $n=23$ remained open. We settle this last case. We prove that for every prime $n\equiv3\pmod4$, in particular for $n=23$, the group of projective symmetries of a geometric $(n_4)$ configuration has order at most four, which explains why $n=23$ resisted the symmetric constructions that settled the other values. Reducing the case of a Klein four-group of symmetries to a few thousand bilinear polynomial systems in $18$ unknowns, and solving all of them exactly with Gröbner bases and saturation, we show that exactly two geometric $(23_4)$ configurations reach this bound, up to projective equivalence, and that there are no others even over $\mathbb{C}$. Both are self-polar with respect to a conic. The first has integer homogeneous coordinates; the second is defined over $\mathbb{Q}(\sqrt{17})$, but not over $\mathbb{Q}$. Consequently geometric $(n_4)$ configurations exist if and only if $n\ge18$ and $n\ne19$. A third configuration, self-polar and defined over a real cubic field, has only a single symmetry of order two. The same classification at $n=22$ and $n=26$ recovers Cuntz's configurations and finds three new combinatorial types at $n=22$ and $94$ at $n=26$, among them a $(26_4)$ configuration that is self-polar with respect to a conic without real points.

We study a model of a random centrally symmetric polytope and ask about the existence of points in the polytope with non-zero, integer coordinates. Our main results are the location of a sharp threshold for the existence of these integer points, a characterization of the typical empirical distribution of their coordinates, and showing that near the feasibility threshold these integer points are almost always totally isolated — at linear Hamming distance to the nearest other integer point in the polytope. The behavior of these random integer programs is related to that of the symmetric binary perceptron (the case when coordinates of points are $\pm 1$-valued). The lack of symmetry in the set of potential solutions leads to new questions and phenomena, such as the distribution of the coordinate sizes in typical solutions, and requires the development of new information-theoretic techniques to deal with higher-dimensional optimization problems arising in the method of moments.

A stable matching in a bipartite preference system is a matching that has no unmatched pair whose endpoints both prefer each other to their current assignments. Rothblum gave a linear description of the stable marriage polytope, the convex hull of the incidence vectors of stable matchings. We prove that this polytope is box-totally dual integral. The proof uses the affine representation of the stable marriage polytope as an order polytope together with the matricial characterization of box-TDI polyhedra. Since Kiraly and Pap proved that Rothblum's description is totally dual integral, our result further implies that Rothblum's system is box-totally dual integral.

The linear arrangement of a simple graph $G = (V,E)$ is a bijective labeling $\pi$ of the vertices of $G$ to the set $\{1, \ldots, n\}$. The maximum linear arrangement ($\texttt{maxLA}$) problem is to determine whether $G$ has a labeling such that the sum of labels across $G$'s edges, $\sum_{uv \in E} |\pi(u) - \pi(v)|$, is at least a given value $k$. Minimum linear arrangement ($\texttt{minLA}$) is defined analogously. Both $\texttt{maxLA}$ and $\texttt{minLA}$ are known to be NP complete for general graphs. $\texttt{minLA}$ has shown to be NP complete for bipartite graphs by Evans and Shiloah in 1975. It has been an open problem whether the same holds for $\texttt{maxLA}$. In this paper we prove that $\texttt{maxLA}$ is NP complete for bipartite graphs.

We study two problems: Local Leakage Resilience (LLR) for Shamir secret sharing, and worst-case Optimal Polynomial Intersection (OPI). Both problems concern polynomials $Q(X)$ of degree less than $k$, over a prime-order finite field $\mathbb{F}_p$. In LLR for Shamir secret sharing, one asks how much one can learn about $Q(0)$ given a few bits leaked from each of $Q(\alpha_1), \ldots, Q(\alpha_n)$, for distinct non-zero evaluation points $\alpha_i \in \mathbb{F}_p$. In OPI, one is given input list $S_1, \ldots, S_n \subset \mathbb{F}_p$, and wants to find a polynomial $Q(X)$ of degree less than $k$ so that $Q(\alpha_i) \in S_i$ for as many $i$ as possible. Leveraging recent connection between these two problems due to (Sun, Wootters 2026), we improve the state-of-the-art for both problems. For LLR, we show that there is some constant $\delta > 0$ so that, as long as $R := k/n \geq 1/2 - \delta$, Shamir secret-sharing is one-bit locally leakage resilient (meaning that one can learn only a negligible amount about $Q(0)$). This is the first result to break the so-called "one-half barrier" for LLR, and improves over the previous best known result, requiring $R \geq 0.668$ (Kasser, 2025). For OPI, we give a quantum algorithm that finds a polynomial $Q(X)$ that agrees with at least a $\mathsf{SCL}_\rho(R)-\varepsilon$ fraction of the lists in expectation, for every fixed $\varepsilon>0$, where $\mathsf{SCL}_\rho$ is the semicircle law of (Jordan et al., 2025). This improves previous algorithmic (and existential) results of (Jo, 2026) and (Horinaga, Yamakawa, 2026). We also give further improved existential results. We also adapt the hardness result of (Yamakawa, Zhandry, 2024) to apply to OPI (rather than a folded version); over large fields, this gives an unconditional separation between the quantum and classical hardness of OPI relative to a membership oracle.

We prove the lower bounds $\mathrm{R}(6,8)\ge 135$ and $\mathrm{R}(8,10)\ge 345$, improving the bounds 134 and 343 listed in the April 2026 revision of Radziszowski's dynamic survey. We give explicit red/blue colorings of $K_{134}$ with no red $K_6$ or blue $K_8$, and of $K_{344}$ with no red $K_8$ or blue $K_{10}$, and verify them with two independently written exhaustive clique checkers. Starting from published colorings, we find these witnesses by adding vertices and repairing the resulting monochromatic cliques through local search. The search uses exact conflict counts, preparation moves aimed at making remaining cliques cheaper to break, and mutations followed by repair. A loss based on the largest monochromatic clique containing each edge yielded a useful intermediate state for the $\mathrm{R}(6,8)$ construction. Guided by the author, an AI coding agent wrote and ran the search code; we describe the methods and document the ancestry of the resulting colorings.

Dualizing monotone Boolean functions (or equivalently, enumerating minimal transversals in hypergraphs) is a long-standing problem whose output-polynomial-time solvability remains open. While various special cases have been extensively studied, the state-of-the-art algorithm for the general case, due to Fredman and Khachiyan, runs in quasipolynomial time. This paper presents a subexponential-time reduction from 3SAT to the complement of Dual: Given a 3CNF formula with $n$ variables, the reduction constructs hypergraphs $H$ and $L$ of total size $2^{O(n^{2/3}(\log n)^{1/3})}$ such that $L \subseteq Tr(H)$ and the formula is satisfiable if and only if $L\neq Tr(H)$, where $Tr(H)$ is the set of minimal transversals of $H$. As a consequence of this reduction, assuming the Exponential Time Hypothesis (ETH), neither Dual nor Dualization admits an algorithm running in $N^{o(\sqrt{\log N/\log\log N})}$ time, where $N$ is the input size for Dual and the combined input and output size for Dualization. In particular, Dualization cannot be solved in output-polynomial time under ETH.

A rational polyhedron has Chvátal rank at most one if a single round of Chvátal-Gomory cuts yields its integer hull. For such polyhedra, integer feasibility is in NP $\cap$ coNP by a result of Boyd and Pulleyblank from the early 1980s, so it is unlikely to be NP-hard. Whether it is polynomial has remained open since then. We answer this question negatively, under either of two standard assumptions from lattice-based cryptography. First, a polynomial-time algorithm for this problem would solve bounded distance decoding with polynomial factors in deterministic polynomial time, contradicting a widely believed conjecture. Second, assuming the hardness of learning with errors, an average-case analogue of bounded distance decoding, the problem is also hard on average, for an efficiently samplable distribution of polytopes. Both results rest on an elementary sufficient condition: a polyhedron has Chvátal rank at most one if its width is less than one along every row of some unimodular matrix. For our polytopes, such a matrix exists but is hard to find.

Norin and Steiner recently proved the Linear Hadwiger Conjecture. That is, there is an absolute constant $C$ such that every graph $G$ satisfies $\chi(G)\leqslant C\,\text{had}(G)$, where $\chi(G)$ is the chromatic number and $\text{had}(G)$ is the Hadwiger number of $G$. Their proof gives a non-optimised constant $C$ of order $10^{100}$. This paper uses extensive AI-based optimisation to show that every graph $G$ satisfies $\chi(G) \leqslant 19{,}885{,}160\,\text{had}(G)$.

The Weisfeiler-Leman (WL) algorithm tests graph isomorphism by assigning colours to $d$-tuples of vertices and iteratively refining the colouring according to neighbourhood configurations until it stabilizes. Increasing $d$ allows the test to detect finer structural differences between graphs, but the computational cost grows exponentially with $d$. The central question is which dimension allows us to distinguish nonisomorphic graphs. We relate a sufficient such dimension to the structure of a graph $G$ through a rank decomposition over $\mathbb F_2$. The two parameters governing the bound are the maximum cut rank $w$ and the fork load $\ell$, which is the largest sum of the parent and two child cut ranks at a fork in the decomposition tree. We prove that WL in dimension $\max\{\ell+1,2w+2\}$ distinguishes every finite nonempty vertex-coloured graph $G$ with such a decomposition from every graph not isomorphic to $G$. Consequently, for $k\geq1$, graphs of rank width at most $k$ can be distinguished from every nonisomorphic graph in dimension $3k+1$, graphs of linear rank width at most $k$ in dimension $2k+2$, and edgeless graphs in dimension one. Uncoloured Cai-Fürer-Immerman (CFI) constructions give linear lower bounds for both width parameters, so the upper bounds are tight up to constant factors.

A graph is $H$-free if it has no induced subgraph isomorphic to $H$. In 2020, Chudnovsky, Goedgebeur, Schaudt, and Zhong characterized all graphs $H$ such that there are only finitely many minimal obstructions to $3$-colouring $H$-free graphs. In general, the minimal obstructions to $k$-colouring $H$-free graphs are the $(k+1)$-vertex-critical $H$-free graphs, those are, the $H$-free graphs $G$ with $\chi(G)=k+1$ but $\chi(G-v)=k$ for every vertex in $G$. In this paper we complete the characterization for all $k > 4$ by showing that there are onky finitely $k$-vertex-critical $H$-free graphs if and only if $H$ is an induced subgraph of $P_4+\ell P_1$ for some $\ell \geq 0$.

This is a Corrigendum and Addendum to Algorithmica paper (2026) 88:51.

We study approximately uniform sampling of satisfying assignments from random $k$-SAT formulas. For every sufficiently large $k$ and density $0 < \alpha \le 2^k/k^{16}$, we prove that, with high probability over the formula, there is a sampler whose output distribution is within total variation distance $\varepsilon$ of the uniform distribution on satisfying assignments and whose expected running time is at most $(nk(\alpha+1)/\varepsilon)^C$, for a universal constant $C$. Our algorithm improves the counting and sampling algorithms obtained by Chen, Lonkar, Wang, Yang, and Yin (STOC 2025) at the density $2^k/\operatorname{poly}(k)$ with running time $(n/\varepsilon)^{\operatorname{poly}(k,\alpha)}$. Our result achieves this density region for sampling with a polynomial degree independent of both the width and the density. Our algorithm separates a high-degree core from the remaining variables, and combines a recursive sampler for the residual formulas with approximate block heat-bath updates on the core. We adapt the recursive insertion-chain framework of Jain, Mizgerd, and Pham (2026) from $2$-trees to ordinary connected violation sets. Expansion and random literal signs yield uniform moment bounds for the resulting correlated lists across all residual formulas, allowing the signed-flow analysis to give a universal polynomial running-time degree. A polymer expansion and an exploration bound establish a polynomial spectral gap for the core dynamics.

The undirected multiple-unicast conjecture [LL04] asserts that network coding offers no throughput advantage over multicommodity flow. We refute this conjecture by constructing a deterministic linear network code over $\mathbb{F}_9$ on a 182-vertex bipartite subgraph of the point-line incidence graph of $\mathrm{PG}(2,9)$. The construction supports $157$ independent unicast sessions at common coding rate at least $1$, while every fractional multicommodity flow has common rate at most $147/157$. By the amplification theorem of~[BGS17], this yields a family of undirected multiple-unicast instances with coding gap $\Omega((\log n)^\varepsilon)$ for some $\varepsilon>0$. We also introduce a nondeterministic model of network coding based on locally verifiable certificates, which guides our construction and may be of independent interest. Building on the high-girth graph and error-correcting code framework of [BH25], we use GPT-6 to find a nondeterministic counterexample based on a new choice of Reed–Solomon local codes, and then convert this example into a causal code using an edge orientation and local search.

Start with the number $2$. At each move, combine two numbers already made, but the operations must be used in the fixed repeating order $+,\times,-,\div$. We ask for the fewest moves needed to make every integer from $1$ to $N$. Since $2$ is already one of the numbers we want and each move makes at most one new number, at least $N-1$ moves are needed. We show that $N-1$ moves are also enough for every $N\ge 9$. Conventional induction cannot work, because a division that comes right after a completed interval $\{1,\dots,P\}$ produces numbers already made. Instead we extend a completed interval $\{1,\dots,P\}$ to $\{1,\dots,3P\}$ all at once, which counting shows is the smallest multiplicative extension $P\to kP$ that can work, and then adjust the last few moves to reach every other $N$. For $9\le N\le 33$ we give explicit sequences, found by computer search.

A perfect Italian dominating function assigns a label from $\{0,1,2\}$ to each vertex so that the labels in the neighborhood of every zero-labeled vertex sum to exactly two. Although the associated decision problem (PID) is NP-complete on chordal graphs, the complexity status on split graphs remains open. We give an $O(n^4)$ time algorithm for split graphs on $n$ vertices. We note that this is in contrast to Roman, Italian, and perfect Roman domination, which are NP-complete on split graphs. We further show that this tractability boundary is tight: PID is NP-hard on graphs at deletion distance one from split graphs. We also provide dichotomy results for PID on restricted extensions of split graphs formed by attaching pendant stars, and establish inapproximability results for the same.

For every fixed proper minor-closed class $\mathscr C$ and every $\epsilon>0$, we give a deterministic LOCAL algorithm that returns a dominating set of size at most $(2a(\mathscr C)+1+\epsilon)\gamma_f(G)$ on every $G\in\mathscr C$. Here $a(\mathscr C)$ is the supremum edge-to-vertex ratio in $\mathscr C$, and $\gamma_f(G)$ is the fractional domination number. The class also admits a deterministic $(1+\epsilon)$-approximation for fractional dominating set and a randomized algorithm that always returns a dominating set and has expected size at most $(1+\epsilon)\gamma(G)$. In each case, the number of rounds depends only on $\mathscr C$ and $\epsilon$. None of these algorithms requires the number of vertices or the maximum degree as part of the input. For planar graphs, this gives the deterministic guarantee $(7+\epsilon)\gamma_f(G)$. Together with the lower bound of Hilke, Lenzen and Suomela, it determines the infimum of the deterministic constant-round approximation ratios for planar minimum dominating set as $7$, settling a question that had remained open since their work. The corresponding infima, measured against the integral optimum, are $7$ for graphs of Euler genus at most any fixed $g\ge0$, $2t-3$ for $K_t$-minor-free graphs with $3\le t\le9$, and $2r+1$ for graphs of treewidth or pathwidth at most any fixed $r\ge1$. We also prove that, for every integer $r\ge1$, no deterministic constant-round LOCAL algorithm achieves an approximation ratio below $2r+1$ on the $r$-th powers of paths, even when every vertex knows the number of vertices. This gives a new proof that the limiting constants are optimal for planar graphs, graphs of bounded treewidth or pathwidth, and $K_t$-minor-free graphs with $3\le t\le9$. For triangle-free planar graphs, the corresponding infimum is $5$.

The switch chain is a Markov chain defined on the set of labelled realizations of a given graphical degree sequence. At each step, a pair of vertex-disjoint edges is chosen at random and the process attempts to replace them with a uniformly chosen perfect matching of the same four vertices, rejecting any proposal that would create a multiple edge. The resulting process is reversible with respect to the uniform distribution on all realizations. We investigate the mixing time of this chain by viewing realizations as the facets of a simplicial complex and studying a variant of the original process called the simplicial switch chain, which we analyze using tools from the theory of high-dimensional expansion. Our technical contributions include a proof that links of faces of sufficiently high codimension are strong spectral expanders and a comparison between the Dirichlet energies of large block updates and two-edge updates. Our main result is an $O(\Delta^{2}m\log m)$ bound on the mixing time of both simplicial and classical switch chains whenever $m\ge C\Delta^{8}$, where $m$ is the number of edges, $\Delta$ is the maximum prescribed degree, and $C>0$ is an absolute constant. For sequences on $n$ vertices with fixed maximum degree, this proves that the chain mixes in $O(n\log n)$ steps, resolving a longstanding conjecture of Cooper, Dyer, and Greenhill and extending its conclusion to irregular degree sequences.

We prove that a random 3-SAT formula with $n$ variables and $\lfloor 4.268n\rfloor$ clauses is unsatisfiable with high probability. The satisfiability threshold has attracted considerable attention: successive upper bounds reached 4.4898 in the work of Díaz, Kirousis, Mitsche, and Pérez-Giménez, while the cavity method predicts a value near 4.267. Our bound brings the rigorous upper estimate close to this prediction. The proof builds on earlier interpolation methods, including the energetic approach of Achlioptas and Menchaca-Méndez. We study the smallest number of clauses that any assignment must violate, comparing the random formula with a simpler system of constraints on individual variables. This comparison reduces the bound to a finite calculation, which we verify using exact rational arithmetic. The proof is formalized in Lean 4.

Dijkstra's algorithm is traditionally classified as a greedy algorithm. We argue that this classification misrepresents the mechanism by which the algorithm determines shortest paths. We present a direct global-to-local proof of correctness. At each iteration, the algorithm identifies a shortest path over the entire set of paths whose endpoints remain unsolved. This global minimum is necessarily a shortest path to its own endpoint. More fundamentally, the algorithm uses already determined shortest paths to establish one additional shortest path at each iteration. We identify this progression as dynamic programming. A three-step exclusion proof shows how the full path space is reduced to frontier candidates, how relaxation removes inferior alternatives, and why minimum extraction yields the global minimum. The choice is thus determined by a global comparison, not by an optimistic local greedy choice. We conclude that the algorithm is more accurately characterized by dynamic programming and deterministic path exclusion than by the traditional greedy classification.

The quickest path reliability problem is to compute the probability that a given flow is transmitted along a single source-destination path within a given time limit. When a budget is imposed, path feasibility is jointly dependent on lead time, transmission cost, and capacity. Existing exact methods may require enumeration of all minimal paths a priori or examination of many partial paths that cannot be completed feasibly. In this paper, we propose an exact algorithm that generates the corresponding lower boundary state vectors without first enumerating minimal paths. Shortest-lead-time, least-cost, and widest-capacity information yield admissible destination-based bounds. Stronger capacity-aware bounds are derived from capacity-filtered subnetworks. These bounds allow early pruning of infeasible branches while keeping all solutions. The algorithm generates the exact complete set of lower bounds without any duplicates, and its correctness and computational complexity are proven.

In neutral-atom quantum computers, atoms are moved to target positions along paths of empty positions, and a target position may be reserved for one species of atom. Motivated by this task, we introduce Single-Move Labeled Token Routing: every source and every target vertex of a graph is assigned a set of labels, and tokens occupy the sources. A solution consists of a matching that assigns each source to a compatible target (one whose label set intersects its own), a route for each matched pair, and a movement order in which, when a token is moved, its route contains no other token. The problem is known to be polynomial-time solvable when every source is compatible with every target, and $\mathsf{NP}$-complete on grid graphs when each source is compatible with exactly one target. We prove that the latter case remains $\mathsf{NP}$-complete on grids and on planar graphs of maximum degree four even when some solution has pairwise edge-disjoint routes. On trees, the problem is known to be $\mathsf{NP}$-complete even for maximum degree three. We study trees through the solution edge multiplicity, the largest number of routes of a solution sharing an edge, and the candidate edge multiplicity, the largest number of compatible pairs whose paths share an edge. We prove that on trees of maximum degree three, the problem is $\mathsf{W}[1]$-hard parameterized by a bound on the solution edge multiplicity, even when a movement order is given, and that on trees of unbounded degree, it is $\mathsf{NP}$-complete even when the candidate edge multiplicity is at most eight. We show that on trees the problem is fixed-parameter tractable parameterized by the maximum degree together with the candidate edge multiplicity, and also by the candidate vertex multiplicity, the same count at vertices. Unless $\mathsf{P}=\mathsf{NP}$, neither the maximum degree nor the candidate edge multiplicity can be omitted.