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Nearly tight bounds for MaxCut in hypergraphs

Oliver Janzer, Julien Portier

TL;DR

This work advances MaxCut theory for hypergraphs by proving an approximate $m^{2/3}$-type surplus bound for all $k$-uniform hypergraphs with large $k$ and all $2\le r\le k$, while achieving a tight $\Omega(m^{3/4})$ surplus for linear hypergraphs. The authors develop a novel modular SDP vector construction that preserves useful correlations while mitigating high codegree complications, and they extend the SDP framework to general hypergraphs via careful reductions to $2$-cuts in auxiliary structures. Complementary constructions, including Steiner-system-based examples, demonstrate the near-tightness of the bounds and illuminate the role of structure in achieving large surpluses. The results have potential implications for related discrepancy problems, bisection width, and positive discrepancy, and the modular-SDP approach offers a versatile toolkit for hypergraph optimization problems.

Abstract

An $r$-cut of a $k$-uniform hypergraph is a partition of its vertex set into $r$ parts, and the size of the cut is the number of edges which have at least one vertex in each part. The study of the possible size of the largest $r$-cut in a $k$-uniform hypergraph was initiated by Erdős and Kleitman in 1968. For graphs, a celebrated result of Edwards states that every $m$-edge graph has a $2$-cut of size $m/2+Ω(m^{1/2})$, which is sharp. In other words, there exists a cut which exceeds the expected size of a random cut by the order of $m^{1/2}$. Conlon, Fox, Kwan and Sudakov proved that any $k$-uniform hypergraph with $m$ edges has an $r$-cut whose size is $Ω(m^{5/9})$ larger than the expected size of a random $r$-cut, provided that $k \geq 4$ or $r \geq 3$. They further conjectured that this can be improved to $Ω(m^{2/3})$, which would be sharp. Recently, Räty and Tomon improved the bound $m^{5/9}$ to $m^{3/5-o(1)}$ when $r \in \{ k-1,k\}$. Using a novel approach, we prove the following approximate version of the Conlon-Fox-Kwan-Sudakov conjecture: for each $\varepsilon>0$, there is some $k_0=k_0(\varepsilon)$ such that for all $k>k_0$ and $2\leq r\leq k$, in every $k$-uniform hypergraph with $m$ edges there exists an $r$-cut exceeding the random one by $Ω(m^{2/3-\varepsilon})$. Moreover, we show that (if $k\geq 4$ or $r\geq 3$) every $k$-uniform linear hypergraph has an $r$-cut exceeding the random one by $Ω(m^{3/4})$, which is tight and proves a conjecture of Räty and Tomon.

Nearly tight bounds for MaxCut in hypergraphs

TL;DR

This work advances MaxCut theory for hypergraphs by proving an approximate -type surplus bound for all -uniform hypergraphs with large and all , while achieving a tight surplus for linear hypergraphs. The authors develop a novel modular SDP vector construction that preserves useful correlations while mitigating high codegree complications, and they extend the SDP framework to general hypergraphs via careful reductions to -cuts in auxiliary structures. Complementary constructions, including Steiner-system-based examples, demonstrate the near-tightness of the bounds and illuminate the role of structure in achieving large surpluses. The results have potential implications for related discrepancy problems, bisection width, and positive discrepancy, and the modular-SDP approach offers a versatile toolkit for hypergraph optimization problems.

Abstract

An -cut of a -uniform hypergraph is a partition of its vertex set into parts, and the size of the cut is the number of edges which have at least one vertex in each part. The study of the possible size of the largest -cut in a -uniform hypergraph was initiated by Erdős and Kleitman in 1968. For graphs, a celebrated result of Edwards states that every -edge graph has a -cut of size , which is sharp. In other words, there exists a cut which exceeds the expected size of a random cut by the order of . Conlon, Fox, Kwan and Sudakov proved that any -uniform hypergraph with edges has an -cut whose size is larger than the expected size of a random -cut, provided that or . They further conjectured that this can be improved to , which would be sharp. Recently, Räty and Tomon improved the bound to when . Using a novel approach, we prove the following approximate version of the Conlon-Fox-Kwan-Sudakov conjecture: for each , there is some such that for all and , in every -uniform hypergraph with edges there exists an -cut exceeding the random one by . Moreover, we show that (if or ) every -uniform linear hypergraph has an -cut exceeding the random one by , which is tight and proves a conjecture of Räty and Tomon.

Paper Structure

This paper contains 21 sections, 48 theorems, 145 equations.

Table of Contents

  1. Introduction
  2. Proof ideas
  3. The basics of our approach
  4. First main idea: a modular definition of the SDP vectors
  5. Second main idea: an SDP bound for $2$-cuts in hypergraphs
  6. Organization of this paper
  7. Preliminaries
  8. An SDP bound for $2$-cut in hypergraphs
  9. Linear hypergraphs: tight lower bound
  10. Upper bound for the surplus of linear hypergraphs
  11. General hypergraphs
  12. The case of small $r$
  13. The case of large $r$
  14. The proof of Lemma \ref{['lem:Correlation-Modulo-Constraint']}
  15. A new construction for $2$-cut in $4$-graphs
  16. ...and 6 more sections

Key Result

Theorem 1.1

For every $2 \leq r \leq k$ with $(r,k) \neq (2,2)$ and $(r,k) \neq (2,3)$, every $k$-uniform hypergraph with $m$ edges has an $r$-cut of surplus $\Omega(m^{5/9})$.

Theorems & Definitions (88)

  • Theorem 1.1: Conlon--Fox--Kwan--Sudakov conlon2019hypergraph
  • Conjecture 1.2: Conlon--Fox--Kwan--Sudakov conlon2019hypergraph
  • Theorem 1.3: Räty--Tomon raty2024large
  • Theorem 1.4
  • Theorem 1.5
  • Proposition 1.6
  • Proposition 1.6
  • Theorem 2.1
  • Lemma 3.1: Azuma-Hoeffding inequality
  • Lemma 3.2
  • ...and 78 more