arXiv subjects

Andrey Kupavskii

Publications and source records attributed to Andrey Kupavskii.

At least 19 recordsLinked to original sources

arXiv2026

A polynomial method over formal power series

We give new upper bounds on families of integer vectors with prescribed coordinate multiplicities and a forbidden scalar product. Our method uses coefficient functions over formal power series and recovers the Frankl--Wilson bound for binary vectors. For vectors in $\{-1,0,1\}^n$, we obtain an exponential improvement over the ordinary polynomial count in a wide range; in certain regimes, we determine the lower limit of the exponential base. We use the new bounds to make progress on several questions in combinatorial geometry. In particular, we improve the base of the exponent in Borsuk's partition problem: we construct finite subsets of $\mathbb R^N$ requiring at least $(1.37+o(1))^{\sqrt N}$ parts of smaller diameter for all sufficiently large $N$, and obtain a base of $1.3839$ along a subsequence of dimensions.

math.CO↗

arXiv2026

Polynomial and Hyperplane Avoidance over the Integers with Applications to Integer-Point Counting

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.

cs.CC↗

arXiv2026

Between the Semi-Random Process and G(n,p)

We study a random graph process that connects the Erdős--Rényi random graph with the semi-random process. In each round, a uniformly random vertex is revealed together with a uniformly random candidate set of $s$ possible partners, and Builder chooses one of them. We study how this limited adaptivity $s$ changes the hitting times of monotone graph properties and how the familiar logarithmic ``coupon--collector'' delays in random graphs are reflected in this setting.

math.CO↗

arXiv2026

Almost entropic matroids and $(k,L)$-systems

For an integer $k$ and a set of integers $L$, a $(k,L)$-system is a family of $k$-element sets such that the size of the intersection of any two distinct members belongs to $L$. Let $m(n,k,L)$ denote the maximum size of a $(k,L)$-system on a ground set of $n$ elements. In this work, we establish a close connection between the asymptotic growth of $m(n,k,L)$ and almost entropic matroids. As a consequence, we obtain a general computable upper bound on the asymptotics of $m(n,k,L)$ that unifies and strengthens all previously known upper bounds.

math.CO↗

arXiv2026

Families that intersect in an arithmetic progression

For $n\in\mathbb N$, let $μ_n$ be the uniform measure on $2^{[n]}$. A family $\mathcal F \subseteq2^{[n]}$ is called $k$-AP-intersecting if $F_1\cap F_2$ contains a nonconstant $k$-term arithmetic progression for every $F_1, F_2\in\mathcal F$. Writing $m_k$ for the supremum of $μ_n(\mathcal F)$ over all $n$ and all such families $\mathcal F$, we prove \[ 2^{-k}\le m_k\le32\cdot 2^{-k}\qquad(k\ge3). \] This greatly improves a recent result of Gillott, who proved that $m_k \leq k^{-1/3}$ for all sufficiently large $k$, and answers his question on $m_k$'s exponential decay. We also prove a similar bound for the corresponding cross-intersection problem.

math.CO↗

arXiv2026

Improved Approximations for Coflow Scheduling via Iterated Rounding and Stable Matchings

We study Coflow Scheduling, introduced by Chowdhury, Zhong, and Stoica (2014). We give deterministic polynomial-time algorithms achieving a $2.77$-approximation without release dates and a $3.17$-approximation with integer release dates. These improve the $3.42$ and $4.36$ guarantees of Rohwedder and Schnaars (2025). Our approach combines two ideas: (1) a novel iterated rounding scheme, and (2) Galvin's list edge coloring method, which is based on the Gale--Shapley stable matching algorithm.

cs.DS↗

arXiv2026

Approximation algorithms via discrepancy with hard constraints

In this paper, we present a discrepancy-based approach to cost-preserving rounding under hard constraints. Starting from a feasible fractional solution, our rounding procedure preserves the designated hard constraints, never increases the linear objective, and bounds the error in the remaining constraints by the discrepancy of a suitable column-difference matrix. Our second discrepancy-theoretic result bounds the discrepancy of a matrix in terms of a novel column parameter, called mountain distance. This notion captures ordered structural features of a matrix which are invisible to the classical Beck--Fiala and Komlós column-norm parameters. We show a variety of applications of these two results. In particular, we obtain new results for min-cost unsplittable flows, bounded-turn routing in grids and tori, rainbow prefix hypergraph decomposition, and $2$-multicast scheduling.

cs.DS↗

arXiv2026

SP-Allreduce: Symmetric Tree Schedules for Tori and Beyond

Allreduce is a fundamental communication primitive in large-scale distributed training, and torus interconnects remain widely deployed in high-performance systems. We introduce SP-allreduce, a new allreduce algorithm for $d$-dimensional tori that exploits the symmetry of the topology and routes only along shortest paths. We call an allreduce algorithm \emph{fully bandwidth-optimal} if it incurs no bandwidth overhead, no multihop-induced traffic inflation, and achieves perfect per-round load balance. SP-allreduce is based on a family of carefully symmetrized shortest-path trees that guarantees full bandwidth optimality in the $α$--$β$ model. We prove that it uses the minimum possible number of rounds among all fully bandwidth-optimal allreduce algorithms on $d$-dimensional tori, and therefore the minimum running time in this class. In this sense, SP-allreduce refines Bucket allreduce: it preserves Bucket's key strengths while reducing the number of rounds by a factor of two. We further extend our approach to broader symmetric topologies and to communication trees of prescribed depths. Experimentally, the resulting algorithms improve over the best compared state-of-the-art algorithms for tori by up to $1.19\times$ on square tori and $1.65\times$ on rectangular tori across the tested configurations. On HyperX and spine-leaf networks, we demonstrate gains of up to $14.82\times$ and $1.72\times$, respectively, over the best evaluated baselines.

cs.DC↗

arXiv2026

Ordinary and measurable colorings of Euclidean three-space

A proper coloring of Euclidean space is a coloring that assigns different colors to points at distance one. Using the planar character-law rigidity theorem of OpenAI, we prove that every finite proper coloring of $\R^3$ can be replaced by a proper Lebesgue-measurable coloring with the same number of colors. As a corollary we obtain that the ordinary and measurable chromatic numbers of $\R^3$ are equal, and both lie between $7$ and $15$.

math.CO↗

arXiv2026

Near-optimal edge partitioning via intersecting families

We study the problem of edge partitioning, where the goal is to partition the edge set of a graph into $k$ parts. The replication factor of a vertex $v$ is the number of parts that contain edges incident to $v$. The goal is to minimize the average replication factor of the vertices while keeping the sizes of the parts nearly equal. We study the regime where the number of parts is significantly smaller than the size of the graph. To this end, we prove asymptotically tight bounds on the optimal replication factor, both for any constant number of parts $k$ and when $k$ grows slowly with the number of vertices. In particular, for growing $k$, every graph admits an almost balanced partition with average replication factor $\sqrt{k}(1+o(1))$, and this bound is tight. The upper bounds are achieved by a new class of edge partitioning algorithms. These algorithms are computationally efficient, including in the LOCAL and CONGEST models, and can be implemented as stateless streaming algorithms in graph processing frameworks. The lower bounds are witnessed by complete graphs and by jumbled graphs, also known as pseudo-random graphs. Our method generalizes a family of algorithms based on symmetric intersecting families of sets. Informally, we replace the symmetry condition by a weaker balance condition that is still sufficient for the algorithms. This relaxation makes it possible to construct such families with asymptotically optimal rank $\sqrt{k}(1+o(1))$.

cs.DM↗

arXiv2026

Nearly Tight Bounds for Proportional Group Fair Divisions and One-Sided Discrepancy

This paper studies the problem of fair division of indivisible goods among $k$ groups of $n_1,\ldots, n_k$ agents. We look at the worst downward deviation $\textit{PROP}(n_1,\ldots, n_k)$ of an agent in a group from its $1/k$-share. We improve the bounds of (Manurangsi and Meka, 2026) and show that $\textit{PROP}(n_1,\ldots, n_k) = \tildeΘ(\sqrt{n/k})$, where $n = n_1 + \ldots + n_k$ is the total number of agents. For the proof of the upper bound, we develop novel discrepancy-type tools and, in particular, a way to efficiently work with one-sided discrepancy constraints.

cs.DM↗

arXiv2026

A Complete Intersection Theorem for Large Permutation Groups

A family of permutations is called $t$-intersecting if any two permutations in the family agree on at least $t$ elements. We prove that there exists $n_0 \in \mathbb{N}$ such that for any $n>n_0$ and any $1 \leq t \leq n$, the maximum size of a $t$-intersecting family in $S_n$ is obtained by one of the families $\mathcal{F}_{n,t,r}=\{σ\in S_n: |\mathrm{Fixed}(σ) \cap \{1,2,\ldots,t+2r\}|\geq t+r\}$, where $\mathrm{Fixed}(σ)$ is the set of fixed points of $σ$. This proves an analogue of the classical Complete Intersection Theorem for large permutation groups, thus providing an essentially complete solution of the Deza-Frankl intersection problem for permutations (1977).

math.CO↗

arXiv2026

Structure and properties of large cross-intersecting families

The study of intersecting families, initiated by Erdős, Ko, and Rado, is a central topic in extremal combinatorics. A classical stability result of Hilton and Milner determines the largest non-trivial intersecting family, and in subsequent works researchers developed structural stability results via the notion of diversity. In this paper, we study cross-intersecting families. We establish a structural theorem for large cross-intersecting pairs, extending Kupavskii's theorem from intersecting families to the cross-intersecting setting. Our result characterizes extremal cross-intersecting pairs in terms of their diversity parts and maximal cross-intersecting extensions. As corollaries, we obtain cross-intersecting analogues of several classical theorems, including those of Han--Kohayakawa and Huang--Peng. A key ingredient in the proof is a new shifting method, called the $S_{U,V}^{Q}$-shift, which not only preserves global intersection properties but also maintains certain local substructures after shifting. We expect this method to be useful elsewhere, and it is already one of the key tools in establishing a product analogue of the Hilton--Milner theorem.

math.CO↗

arXiv2026

More on the Erd\H os--Kleitman problem on matchings in set families

Let $e(n,s)$ denote the maximum size of a family $\mathcal{F}$ of subsets of an $n$-element set that contains no $s$ pairwise disjoint members. In 1968, answering a question of Erdős, Kleitman determined $e(sm-1,s)$ and $e(sm,s)$ for all integers $m,s\ge 1$. Half a century later, Frankl and Kupavskii determined $e(s(m+1)-\ell, s)$ for $\ell \leq \frac{s-3}{m+3}$. They showed that the corresponding extremal example is closely connected with the extremal example for the Erdős Matching Conjecture, and conjectured that the same remains true for all $\ell \leq s/2$. In this paper, we prove an approximate version of their conjecture for $s\ge s_0(m)$.

math.CO↗

arXiv2026

Matchings in permutations

We say that two permutations $[n]\to [n]$ intersect if they map some element $x$ to the same element $y$. A matching in a family of permutations is a collection of pairwise disjoint permutations. In this paper, we study families of permutations with no matchings of size $s$. In particular, we obtain a characterization of the largest $s$-matching-free families and a Hilton--Milner type result. We also obtain results for the families of derangements.

math.CO↗

arXiv2026

Structure of $t$-Intersecting Families of Vector Spaces

We study $t$-intersecting and $t$-cross-intersecting families of $k$-dimensional subspaces in finite vector spaces of dimension $n$. We show that all large $t$-intersecting families admit a governing low-dimensional structure for $n \ge 2k+1$. This result, together with its cross-intersecting variant, allows us to prove analogues of several classical extremal set-theoretic results. In particular, we determine the intersecting families with the largest diversity, and we establish a Frankl-type degree-diversity result that generalizes the Hilton-Milner theorem. Our proofs rely on simplification procedures for $t$-intersecting and $t$-cross-intersecting families of subspaces. These procedures are based on the concept of subspace spreadness, a generalization of the classical notion of spreadness for set systems.

math.CO↗

arXiv2026

Families without $s$-matchings: the other end

In this paper, we determine the largest family $\mathcal F \subset 2^{[n]}$ without $s$ pairwise disjoint sets, provided $n=ms+c$ for positive integers $m,c$, and $s \geq s_0(m, c)$. This result can be seen as a non-uniform analogue of the results on the Erd\H os Matching Conjecture in the regime when the clique is extremal.

math.CO↗

arXiv2026

Frankl's diversity theorem for permutations

In 1987, Frankl proved an influential stability result for the Erd\H os--Ko--Rado theorem, which bounds the size of an intersecting family in terms of its distance from the nearest (subset of) star or trivial intersecting family. It is a far-reaching extension of the Hilton--Milner theorem. In this paper, we prove its analogue for permutations on $\{1,\ldots, n\}$, provided $n$ is large. This provides a similar extension of a Hilton--Milner type result for permutations proved by Ellis.

math.CO↗