Average pairwise Hamming distance of a subset in a hypercube
Yijia Fang, Hao Huang
AI 速览：作者证明密度为1/8的{0,1}^n子集的平均成对汉明距离至少为(n-3)/2，且等号仅由(n-3)维子立方体达到。
In this note, we prove that for a subset $\mathcal{F}$ of $\{0,1\}^n$ of density $\frac{1}{8}$, its average pairwise Hamming distance is at least $\frac{n-3}{2}$, with equality attained only by $(n-3)$-dimensional subcubes. This answers a question of Lei Yu, and solves an important open case of a more general question posed by Ahlswede and Katona in 1977.
Half-Optimal Induced Matchings in Random Graphs: Online and Low-Degree Algorithms
Abhishek Dhawan, Eren C. Kızıldağ, Neeladri Maitra, Cheng Mao, Timothy L. H. Wee
AI 速览：作者证明在 Erdős-Rényi 随机图中，寻找大诱导匹配存在尖锐的因子二间隙：简单在线贪心可达约一半最优，而在线算法与低阶多项式算法均无法突破该阈值。
For many optimization problems on random graphs, most prominently the largest independent set problem, natural algorithms reach only half the optimum, and such factor-two gaps are known to be sharp within broad algorithmic classes. We prove such a gap for finding large induced matchings in Erdős-Rényi random graphs. In the worst case, requiring a matching to be induced turns a polynomial-time solvable problem into an NP-hard one; our results reveal an average-case counterpart. In $G(n,p)$ with $C/n\le p\le 1-κ$, the largest induced matching has size $(1+o(1))L$ with high probability, where $L=\log_{1/(1-p)}(np)$. A simple online greedy algorithm finds an induced matching of size $(1/2-ε)L$, while no online algorithm, even with adaptive inspection order and unbounded computation, finds one of size $(1/2+ε)L$ except with probability $\exp(-Ω(L\log(np)))$. In the sparse regime $p=d/n$, low-degree polynomial algorithms exhibit the same sharp threshold. Our guarantees rest on a maximality principle: with high probability, every maximal induced matching is half-optimal. Greedy outputs a maximal induced matching and our low-degree algorithm a near-maximal one. We bypass the delicate coupling arguments classically used to analyze greedy since Karp. For low-degree hardness, we adapt Wein's overlap gap argument, with forbidden structures that track overlaps between endpoint sets rather than edge sets, since a new edge may join two previously used endpoints and thus encounter no fresh randomness. For online hardness, we give a new and significantly simpler argument that avoids the correlated instances of prior OGP-based proofs: at a suitable stopping time, all pairs involving uninspected vertices remain unrevealed, so a direct first-moment bound suffices.
Recently, Chudnovsky, Cook, Davies, Oum, and Tan obtained the first finite bound on the chromatic number of t-perfect graphs, showing that they are 199053-colorable. We improve this bound to 130 by refining their proof. The original proof establishes that every graph with large odd girth and large chromatic number contains a certain structure called an r-arithmetic rope, and that its existence in a certain leveling of a graph with large odd girth would imply an odd wheel as a t-minor, a known obstruction to t-perfectness. While their technique requires a lower bound on the chromatic number that is exponential in r, we show that the existence of an r-arithmetic rope can already be guaranteed under a linear bound. Using a slightly weakened notion of arithmetic ropes allows us to reduce the bound even further.
Layering and reverse extension for weighted $k$-path vertex cover
Jiajing Wu, An Zhang, Shiming Li, Guohui Lin
AI 速览：作者在分层与反向扩展局部比值框架下设计局部权重函数，为加权 k-路径顶点覆盖问题给出新的近似比。
We study the weighted $k$-path vertex cover problem for any fixed integer $k \ge 2$, in which a vertex non-negatively weighted simple graph $G = (V, E)$ is given and one wants to find a minimum-weight vertex subset $C \subseteq V$ such that every simple path on $k$ vertices in $G$ contains at least one vertex of $C$. The problem generalizes the classic weighted vertex cover problem, that is, $k = 2$, and it admits a $k$-approximation algorithm of Hochbaum~[SIAM J. Comput., 1982] and a $(k - \frac 12)$-approximation algorithm of Brüstle et al.~[IWOCA 2021]. In this paper, we carefully design local weight functions in the Layering and Reverse Extension local ratio framework of Bar-Yehuda et al.~[ACM Comput. Surv., 2004], leading to an approximation algorithm of ratio $\frac {3k-1}4$ for odd $k \ge 3$ and of ratio $\frac {3k}4$ for even $k \ge 4$. When $k = 3$ and $k = 4$, these ratios are $2$ and $3$, respectively, matching the state-of-the-art approximation ratios for the corresponding special cases.
The simple continued fractions for the Golden & Silver means are well-known. It is astonishing that, as far as we know, no one has published half-iterates (let alone quarter-iterates) for the corresponding algorithms. We also examine the cosine and logistic maps (with parameter $2 < λ< 3$).
We complete the determination of the minimum possible length of eight-dimensional binary self-orthogonal codes with given minimum distance. For dimension nine we give exact values and bounds for minimum distance at most $64$. By exploiting the geometric reformulation of linear codes as multisets of points, we fully redo the classification for dimension at most seven in a simplified and computer-free manner. As a by-product we prove the non-existence of $[93,9,44]_2$ codes and show several divisibility results for putative optimal nine-dimensional binary codes.
We prove the following asymptotically tight lower bound for $k$-color discrepancy: For any $k \geq 2$, there exists a hypergraph with $n$ hyperedges such that its $k$-color discrepancy is at least $Ω(\sqrt{n})$. This improves on the previously known lower bound of $Ω(\sqrt{n/\log k})$ due to Caragiannis et al. (arXiv:2502.10516). As an application, we show that our result implies improved lower bounds for group fair division.
We prove that finite two-graphs have coherent EPPA and that neither graphs modulo global complementation nor two-graphs modulo global complementation have EPPA. This completes the classification of EPPA and coherent EPPA for the reducts of the random graph, answering a question and confirming a conjecture of Evans, Hubička, Konečný and Nešetřil.
Yuta Inoue, Ken-ichi Kawarabayashi, Atsuyuki Miyashita, Bojan Mohar, Carsten Thomassen, Mikkel Thorup
The famous Four Color Theorem (4CT) states that any planar graph can be colored with four colors, but how do we find a 4-coloring? In our previous paper [FOCS'26], we presented a simple combinatorial algorithm to 4-color planar graphs in $O(n\log n)$ time. We improve it here to $O(n)$ time, which is optimal. The algorithm from our previous paper is based on proving that every planar triangulation has a linear number of reducible configurations or obstructing cycles. The bottleneck for the algorithm is that we need $O(\log n)$ Kempe changes to extend a recursive 4-coloring to the reducible configurations. Each Kempe change can involve a constant fraction of the graph and takes linear time. Here we show that all $O(\log n)$ Kempe changes can be identified and implemented in $O(n)$ time. The idea is to spend $O(n)$ time to preprocess the graph so that we can identify and simulate each good Kempe change in $O(n/\log n)$ amortized time. Thus, we implement all $O(\log n)$ Kempe changes in $O(n)$ total time. Based on this, we implement the whole 4-coloring in linear time. We note that creating such a representation for fast identification and simulation of the right Kempe changes is a challenge of data structures, unlike the previous paper that builds on a new graph theoretic understanding of reducible configurations.
For an integer $h\geq2$, an edge-ordered $h$-uniform hypergraph $H^<$ is an $h$-uniform hypergraph with a linear ordering of its hyperedges. For an integer $q \geq 1$, the edge-ordered Ramsey number $R_{\mathrm{edge}}(H^<;q)$ of $H^<$ is the smallest positive integer $N$ such that there exists an edge-ordered $h$-uniform hypergraph $G^<$ on $N$ vertices such that, for every $q$-coloring of the hyperedges of $G^<$, there is a monochromatic copy of $H^<$ in $G^<$ preserving the hyperedge order. In this note, we show that, for every edge-ordered $h$-uniform hypergraph $F^<$ on $n$ vertices with $m\geq1$ hyperedges, $R_{\mathrm{edge}}(F^<;q) \leq R_h(F;q)^{Cm^2}$ where $F$ is the underlying unordered hypergraph, $R_h(F;q)$ is its standard $q$-color Ramsey number, and $C=C(h,q)>0$ is a constant. For fixed $h\geq3$ and $q\geq2$, this shows that prescribing the hyperedge order does not require an increase in the tower height of the classical upper bound. It also settles a problem posed by Fox and Li (2020), and for $h=2$ gives the bound $2^{O(n^4\log n)}$, improving their bounds for every fixed $q \geq 3$.
Triadic closure is a natural mechanism for modeling the clustering phenomenon observed in many real-world networks. Given a possibly random undirected graph $H$, we consider pairs of non-adjacent vertices that have a common neighbor. Conditional on $H$, each such pair is joined by an edge independently and simultaneously, with a probability depending on the degrees of its endpoints and on the number of their common neighbors. The resulting graph is called a triadic closure of $H$. We study the effect of this operation on local convergence. More precisely, we prove that if a sequence of potentially random finite undirected graphs of $\left( H_n \right)_{n \in \mathbb{N}}$ converges locally in probability to a rooted graph $H^*$, then the corresponding sequence of triadic closures converges locally in probability to the same triadic closure of $H^*$. As a consequence, we obtain a general limit theorem for the clustering coefficients of triadic closures. In particular, we show that the triadic closures can have a strictly positive limiting clustering coefficients even when the limiting clustering coefficients of the parent graph sequence is zero. Finally, we apply our results to several classical sparse random graph models including Erdős-Rényi, configuration, preferential attachment models and compute the limiting clustering coefficients of their triadic closures.
Expander graphs have been central objects of study for over fifty years. Ramanujan graphs attain the asymptotically optimal nontrivial adjacency-eigenvalue threshold for infinite fixed-degree families. The original arithmetic constructions by Lubotzky-Phillips-Sarnak (1988) and Margulis (1988), and the iterative $2$-lift method of Bilu-Linial (2006) provide different approaches to Ramanujan graph constructions. Motivated by work on Mealy automata, we combine both approaches to give the following new arithmetic construction. For every prime $p\equiv 1\pmod {17}$, we construct a infinite tower of $(p+1)$-regular non-bipartite Ramanujan graphs on $2^n$ vertices, with label-preserving $2$-lifts between consecutive levels. These construction is strongly explicit: for fixed $p$, a neighbor or lift sign is computable in $O_p(n^3)$ bit operations. The construction uses fixed algebraic $2$-adic transformations, specified by finite data computable in preprocessing depending only on $p$. The main number-theoretic tool is Blasius's Ramanujan-Petersson theorem for Hilbert modular forms, applied through the Jacquet-Langlands correspondence to obtain the sharp bound $|λ|\leq2\sqrt p$ for every nontrivial adjacency eigenvalue of our $(p+1)$-regular graphs. The transformations preserve $2$-adic distance, which gives the binary covers and an explicit formula for their signings.
A set $S$ of vertices of a graph $G$ is a mutual-visibility set if every pair of vertices of $S$ is joined by a shortest path whose internal vertices avoid $S$, and the mutual-visibility chromatic number $χ_μ(G)$ is the least number of mutual-visibility sets into which $V(G)$ can be partitioned. Klavžar, Kuziak, Valenzuela Tripodoro and Yero (2025) proved that $χ_μ(G)\leχ(G)$ whenever $G$ has diameter two, observed that equality holds for complete bipartite graphs, and asked whether there are any other diameter-two graphs attaining equality. We answer this question in the affirmative. We construct a graph $G_{11}$ of order 11 and diameter two with $χ_μ(G_{11})=χ(G_{11})=3$, and an infinite family of such graphs obtained from $G_{11}$ by replacing one vertex with pairwise non-adjacent copies of it.
An identifying code of a graph is a dominating set $C$ such that the sets $N[v]\cap C$ are pairwise distinct. The minimum density of an identifying code has been studied extensively for several infinite grids. It is known to be $1/4$ for the triangular grid and $7/20$ for the square grid, whereas for the hexagonal grid the problem is still open, between $5/12$ and $53/126$. These three grids are the regular members of the family of the eleven Archimedean grids. In this paper, we start a systematic study of identifying codes in this family. We determine the minimum density for five of the nine Archimedean grids for which it was not known: it is $2/3$ for the truncated hexagonal grid, $3/7$ for the truncated square grid, and $1/3$ for the trihexagonal (kagome) grid, the elongated triangular grid and the snub hexagonal grid. All lower bounds are proved by the discharging method. For the three remaining non-regular Archimedean grids, we also obtain some bounds: the minimum density lies between $3/7$ and $13/30$ for the truncated trihexagonal grid, between $1/3$ and $11/30$ for the rhombitrihexagonal grid, and between $2/7$ and $9/28$ for the snub square grid. As a by-product, we prove that every identifying code of a cubic graph with the slow growth property in which every vertex lies on exactly one triangle (resp. $4$-cycle) has density at least $2/3$ (resp. $3/7$).
The balance constant is a poset parameter measuring how well random linear extensions can be split according to their relative order on two elements. We give an $\varepsilon$-improvement for the balance constant over the bound by Brightwell, Felsner and Trotter (1995), for some small $\varepsilon>0$. This is the first general result towards the $\frac{1}{3}$--$\frac{2}{3}$ conjecture in over 30 years.
Joshua Brakensiek, David Conlon, Venkatesan Guruswami, Xiaoyu He, Jiaxi Nie, Maya Sankar, Hung-Hsun Hans Yu
In the study of constraint satisfaction problems (CSPs), the non-redundancy of a relation measures the maximum number of clauses in a CSP instance which can be logically independent. In the last few years, CSP non-redundancy has found a number of applications including in query complexity, sparsification, kernelization and streaming. In this paper, we present a novel technique for upper-bounding the non-redundancy of some relations using locally decodable codes (LDCs) and locally correctable codes (LCCs). More precisely, known limitations on the structure of 2-query LDCs and LCCs can be translated into limitations on the size of certain non-redundant CSP instances. This technique builds on a technique used by Brakensiek and Guruswami [STOC 2025] to bound non-redundancy via techniques from matching vector codes. Using LDCs and LCCs, we give improved upper bounds on non-redundancy of the BCK predicate pioneered by Bessiere, Carbonnel, and Katsirelos [AAAI 2020] as well as its well-studied Boolean variant. We also give the first example of a Boolean predicate whose non-redundancy is provably not of the form $n^{k\pm o(1)}$ for some positive integer $k$.
Rigidity and global rigidity are fundamental graph properties in discrete geometry, with applications to structural engineering and network localization. In two dimensions, both admit combinatorial characterizations in terms of the rigidity matroid and graph connectivity. We study the computational complexity of finding minimum-cost sets of edges or vertices whose deletion destroys either property in a finite simple graph. With unit costs, we give deterministic polynomial-time algorithms for edge deletion destroying global rigidity and vertex deletion destroying either property. We prove that edge deletion destroying rigidity is NP-hard and that, with positive integer costs, all four problems are NP-hard. These results resolve, in dimension two, the open questions on recognizing robustness under edge and vertex deletions posed by Jordán and Whiteley (2017), as well as the minimum edge-deletion problem for rigidity recorded in EGRES Open.
Every semi-magic square of order $n$ satisfies the axial moment identities $\sum_{i,j}s_{ij}j^m=\sum_{i,j}s_{ij}i^m=M(n)\sum_jj^m$ for all $m\ge1$, where $M(n)=n(n^2+1)/2$. For real matrices the converse requires $1\le m\le n-1$, whereas for arrangements of $\{1,\dots,n^2\}$ fewer moments suffice once $n\ge14$. We define the permutation threshold $k^*(n)$ as the least $k$ such that every arrangement satisfying these identities for $1\le m\le k$ is semi-magic. In terms of the column-sum deviations $γ_j$, the degree-$k$ conditions state that $\sum_jγ_jt^j$ is divisible by $(t-1)^{k+1}$, linking $k^*(n)$ to the Prouhet--Tarry--Escott problem. While Schur's bound on real zeros gives $k^*(n)\le\sqrt{2(n-1)\log(n^4/4)}$, we prove the converse realisability bound $k^*(n)\ge(\sqrt2-o(1))\sqrt n$ (and $k^*(n)\ge(\sqrt6-o(1))\sqrt n$ when $4\mid n$) by perturbing semi-magic squares with complementary row pairs via horizontal swaps. We also show that $k^*(n)=n-1$ for $3\le n\le13$, $k^*(n)=n-2$ for $14\le n\le17$, and $k^*(n)\le n-3$ for $n\ge18$, and determine $k^*(n)$ for $18\le n\le40$ with computer assistance. Finally, at the $k=1$ level of the hierarchy, we specialise the Mantel--Hubert quadratic assignment variance formula to obtain exact means, variances, and limit laws under a uniform random arrangement for the full line-sum energy $E_{\mathrm{full}}$, for the four-term energy $E_4$ combining the first moments and the two diagonals, and for Loly's moment of inertia, and find that $|L_4|=14{,}639{,}440$ arrangements of order $4$ satisfy $E_4=0$.
Given an arrangement of $\{1,\dots,n^2\}$ in an $n\times n$ grid, we lift each cell to $\mathbb{R}^3$ by centring its grid coordinates $(x,y)$ at the origin and taking its centred entry $z = s_{ij} - (n^2+1)/2$ as a vertical height. The convex hull of these $n^2$ points is the gem of the arrangement, whose lower and upper boundaries are the regular subdivisions of the grid induced by $\pm z$. While random arrangements and generic linear magic squares have gems whose vertex counts grow with $n$, our main theorem shows that the classical Siamese (de la Loubère) magic square of every odd order $n\ge5$ has $f$-vector $(12,26,16)$, isometry group $\{\pm\mathrm{id}\}$, and only the twelve vertices forced by the boundary of the grid. The proof shears the lift onto five horizontal terraces and reduces the hull calculation to a finite table of polynomials in $m=(n-1)/2$; the same method gives $f$-vector $(12,24,14)$ for the base-$n$ digit-exchanged square. Along any family of linear squares with fixed steps the $f$-vector is eventually periodic in $n$, whereas over all linear magic squares of prime order $n$ the maximum vertex count grows as $Θ(\ln n)$. Whenever the heights have vanishing first moments, a short principal-axes argument shows that the point set has a non-trivial isometry if and only if the arrangement is self-complementary up to dihedral symmetry. An exact census of the $880$ magic squares of order four yields $616$ congruence classes and $352$ combinatorial types of gems, and shows that while vertex and facet counts are nearly insensitive to the magic line-sum conditions, the volume and solid centroid shift systematically.
L-extendable functions on the integer lattice are functions that admit a $\frac12$UJ-convex relaxation; that is, values can be assigned to the additional half-integral points, while keeping the original values at integer points, so that the resulting function is $\frac12$UJ-convex. Various examples of L-extendable functions and their applications are known. A general characterization of L-extendability for functions on the integer lattice, however, remains open. In this paper, we study the L-extendability of functions on the integer lattice. We first obtain a necessary condition: L-extendable functions are integrally convex. We then consider a special subclass of L-extendable functions, namely, those whose local convex extensions give L-convex relaxations. This subclass has a useful optimization property: the optimal value of the original function coincides with that of the $\frac12$UJ-convex relaxation given by its local convex extension, a property not guaranteed for general $\frac12$UJ-convex relaxations. We characterize this subclass by showing that a function is L-extendable through its local convex extension if and only if it is integrally convex and its restriction to every unit hypercube is D-convex, where D-convexity is a discrete convexity associated with reflected Freudenthal simplicial divisions. This subclass strictly contains the class of D-convex functions. We also give a combinatorial characterization of D-convex subsets of the vertices of a unit hypercube: a set $X\subseteq\{0,1\}^n$ is D-convex if and only if it is majority-closed and contains a relative opposite pair.
We show that for every prime power $q$ and $ρ\in(0,1-\frac1q)$ there exists $\varepsilon_0(q, ρ) > 0$ such that for all positive $\varepsilon < \varepsilon_0(q, ρ)$, a random linear code over $\mathbb{F}_q$ of rate $1-h_q(ρ)-\varepsilon$ has list size $\lfloor h_q(ρ)/\varepsilon\rfloor$ or $\lfloor h_q(ρ)/\varepsilon\rfloor+1$ with probability $1-o(1)$. Before this work, the list size was known only up to a constant factor for $q\ge3$, and for $q=2$ it was known to be either $\lfloor h_2(ρ)/\varepsilon\rfloor-1$, $\lfloor h_2(ρ)/\varepsilon\rfloor$ or $\lfloor h_2(ρ)/\varepsilon\rfloor+1$. We also show that the same two-point concentration holds for the list size of random LDPC codes. This follows from a general theorem establishing that random LDPC codes and random linear codes have the same threshold rate for distance, list decoding, list recovery from errors and any other \emph{convex} property, which answers a question of Mosheiff, Resch, Ron-Zewi, Silas and Wootters.
Eleon Bach, Yann Disser, Sophie Huiberts, Nils Mosis
We prove that, for every fixed $ε\in(0,1]$, the active-set method requires exponentially many iterations in the worst case to maximize a convex objective of the form $(x_1)^{1+ε}+\vec c^{\top}\vec x$ subject to linear constraints, regardless of the pivot rule used. Our lower bound therefore holds for objectives arbitrarily close to the linear case $ε=0$, where the active-set method coincides with the simplex method and a lower bound for all pivot rules would constitute a major breakthrough. The special case $ε=1$ yields a lower bound in convex quadratic maximization. This improves on the best previously known lower bound [IPCO 2025] requiring polynomial objectives of degree $ω(\log d)$ in dimension $d$ and firmly resolves the open question of whether a constant degree suffices. Our result is based on a novel extended formulation, recursively constructed using deformed products. Its key feature is that it projects onto a polygonal approximation of a parabola while preserving all of its exponentially many vertices. We define a convex objective with a single non-linear variable that forces the active-set method to follow the parabolic boundary of this projection, without allowing any shortcuts along chords corresponding to edges of its full-dimensional preimage.
We analyze the expected running time of WalkSAT, a well-known local search procedure for satisfiability solving, on satisfiable instances of the k-XOR SAT problem. We obtain estimates of this expected running time by reducing the problem to a setting amenable to classical techniques from drift analysis. A crucial ingredient of this reduction is the definition of (new, explosive) hypergraph versions of interacting particle systems, notably of coalescing and annihilating random walks as well as the voter model. The use of these tools allows to show that the expected running time of WalkSAT depends on structural parameter (we call odd Cheeger drift) of the dual of the formula hypergraph.
Let $P$ be a polytope defined by the system $A x \leq b$, where $A \in R^{m \times n}$, $b \in R^m$, and $\text{rank}(A) = n$. We give a short geometric proof of the following tight upper bound on the number of vertices of $P$: $$ n! \cdot \fracΔ{Δ_{\text{average}}} \cdot \text{vol}(B_2) \sim \frac{1}{\sqrt{πn}} \cdot \left(\frac{2 π}{e}\right)^{n/2} \cdot n^{n/2} \cdot \fracΔ{Δ_{\text{average}}}, $$ where $Δ$ is the maximum absolute value of $n \times n$ subdeterminants of $A$, and $Δ_{\text{average}}$ is the average absolute value of subdeterminants of $A$ corresponding to a triangulation of $P$'s normal fan. Assuming that $A$ is integer, such polyhedra are called $Δ$-modular polyhedra. Note that in the integer case, the bound can be simplified via the inequality $Δ_{\text{average}} \geq Δ_{\min} \geq 1$, where $Δ_{\min}$ is the minimum absolute value of subdeterminants of $A$ corresponding to feasible bases of $A x \leq b$. For this, we prove and use a symmetric variant of Macbeath's theorem. Additionally, we give a direct argument based on prior results in the field, showing that the graph diameter of $P$ is bounded by $O\bigl(n^3 \cdot \fracΔ{Δ_{\min}} \cdot \ln (n \fracΔ{Δ_{\min}}) \bigr)$. Thus, both characteristic of $P$ are linear in $Δ/Δ_{\min}$. From an algorithmic perspective, we demonstrate that: Given $A \in Q^{m \times n}$, $b \in Q^m$, and an initial feasible solution to $A x \leq b$, the convex hull of $P$ can be constructed in $O(n)^{n/2} \cdot m^2 \cdot \fracΔ{Δ_{\text{average}}}$ operations. For simple polyhedra, the dependence on $m$ reduces to linear; Given $A \in Z^{m \times n}$ and $b \in Q^m$, the number $|P \cap Z^n|$ can be computed in $O(n)^n \cdot \frac{Δ^4}{Δ_{\text{average}}}$ arithmetic operations.
Local-to-Global theorems are a cornerstone of many of recent developments in theoretical computer science from coding theory and PCPs all the way to analyzing mixing time of Markov chains. Building on trickledown theorems on simplicial complexes, several techniques have been established to obtain tighter local-to-global theorems to analyze the spectral gap of the Glauber dynamics, namely matrix trickledown, ${\mathcal C}$-Lorentzian polynomials and more recently Bochner type inequalities. In this paper, we adopt non-local local-to-global/Bochner type inequalities on "cones", corresponding to the local neighborhood of a given vertex, to show that the Glauber dynamics mixes in polynomial time to sample proper colorings of maximum degree $Δ$ graphs as long as $q\geq 1.7408Δ$.
We count Boolean functions represented as the sign of a real polynomial of bounded degree on the $n$-dimensional cube. We prove that the logarithm of their number is asymptotic to that of the classical hyperplane-arrangement upper bound, uniformly over all degrees, with relative error $O((\log n)^2/n)$. We also obtain sharper additive and relative error estimates in particular degree ranges, including an additive error $O(n^2)$ between the logarithms throughout a small linear degree range and improved bounds just below the central degree. The proof combines probabilistic estimates for Reed--Muller codes with a deletion argument for sets of evaluation vectors, an entropy count, and a sharp-threshold estimate for expected rank deficiency.
Robust optimization models that allow decisions to adapt to newly revealed information remain computationally challenging and are often limited to two decision stages or are addressed heuristically or via approximation schemes. This challenge is further amplified when uncertainty sets depend on earlier decisions. Conventional approaches to such problems often require problem-specific reformulations and tailored algorithms, highlighting the need for an exact model-and-run approach. We address this gap by presenting an algorithmic framework for solving multistage robust discrete linear optimization problems. This exact approach adapts and integrates ideas from game-tree search, quantified conflict-driven clause learning, and integer linear programming to explore the minimax tree and is implemented in the open-source solver Yasol. The solver supports discrete decision variables across multiple stages, discrete uncertainty sets with polyhedral or decision-dependent structure, and mixed-integer recourse in the final stage. It can be used as a standalone executable or through a C++ library interface, allowing it to serve as a reference solver for exact methods in multistage robust discrete linear optimization. In an extensive computational study, we demonstrate the capabilities of our solver. The results show competitive performance against a reference approach from the literature for a protection-interdiction problem, as well as strong performance on bilevel benchmark instances. For a multistage cutting-stock problem, we demonstrate the substantial loss in solution quality that can arise from neglecting the multistage structure and show that our solver significantly outperforms decision-rule approximations in terms of computational efficiency while certifying the true multistage optimum.
We study the \emph{Polynomial Avoidance Problem}: given a finite collection $F \subseteq Z[x_1,\dots,x_n]$ of nonzero polynomials, find a vector $x \in Z^n$ of minimum $\ell_p$-norm such that $f(x)\neq0$ for every $f \in F$. Each polynomial is given sparsely as a list of its nonzero terms. The affine special case is the \emph{Hyperplane Avoidance Problem}. This special case arises in Barvinok-type algorithms for counting integer points in polyhedra, when certain rational generating functions must be evaluated at a removable singularity. Our main results are as follows: 1) Let $D:=\sum_{f\in F} \text{deg} f$. Every instance admits a feasible vector $x$ satisfying \[ \|x\|_1 \leq \min\left\{D,\frac{D+n}{2}\right\}. \] This bound is tight. Such a vector can be found by a deterministic algorithm whose bit complexity is polynomial in $D$ and the sparse encoding length of the input. Our proof uses a new algorithmic variant of the \emph{Combinatorial Nullstellensatz}; 2) In the affine case, with $m$ input polynomials, such a vector can be constructed by a fast deterministic polynomial-time algorithm using $O(nm)$ arithmetic operations. For the homogeneous case, Barvinok's original approach provides only the weaker guarantee $\|x\|_1 = O((nm)^{n-1})$; 3) The problem is NP-hard for every $p\in R_{\geq 1}\cup\{\infty\}$, even for homogeneous linear forms; 4) As an application, we improve the complexity bound for counting integer points in general rational polytopes. We also improve the bound for polyhedra of bounded codimension; this includes the counting version of the \emph{Unbounded Subset-Sum} problem as a special case.