Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Improved Certificates for Independence Number in Semirandom Hypergraphs

Pravesh Kothari, Anand Louis, Rameesh Paul, Prasad Raghavendra

TL;DR

Two key improvements are made: firstly, sharper bounds are proved that get rid of pesky logarithmic factors in $n$, and nearly attain the conjectured optimal computational threshold of $O(\sqrt{n}/p^{1/\ell})$, and secondly, robust Sum-of-Squares (SoS) certificates are designed, proving the authors' bounds in the more challenging semirandom hypergraph model.

Abstract

We study the problem of efficiently certifying upper bounds on the independence number of $\ell$-uniform hypergraphs. This is a notoriously hard problem, with efficient algorithms failing to approximate the independence number within $n^{1-ε}$ factor in the worst case [Has99, Zuc07]. We study the problem in random and semirandom hypergraphs. There is a folklore reduction to the graph case, achieving a certifiable bound of $O(\sqrt{n/p})$. More recently, the work [GKM22] improved this by constructing spectral certificates that yield a bound of $O(\sqrt{n}.\mathrm{polylog}(n)/p^{1/(\ell/2)})$. We make two key improvements: firstly, we prove sharper bounds that get rid of pesky logarithmic factors in $n$, and nearly attain the conjectured optimal (in both $n$ and $p$) computational threshold of $O(\sqrt{n}/p^{1/\ell})$, and secondly, we design robust Sum-of-Squares (SoS) certificates, proving our bounds in the more challenging semirandom hypergraph model. Our analysis employs the proofs-to-algorithms paradigm [BS16, FKP19] in showing an upper bound for pseudo-expectation of degree-$2\ell$ SoS relaxation of the natural polynomial system for maximum independent set. The challenging case is odd-arity hypergraphs, where we employ a tensor-based analysis that reduces the problem to proving bounds on a natural class of random chaos matrices associated with $\ell$-uniform hypergraphs. Previous bounds [AMP21, RT23] have a logarithmic dependence, which we remove by leveraging recent progress on matrix concentration inequalities [BBvH23, BLNvH25]; we believe these may be useful in other hypergraph problems. As an application, we show our improved certificates can be combined with an SoS relaxation of a natural $r$-coloring polynomial system to recover an arbitrary planted $r$-colorable subhypergraph in a semirandom model along the lines of [LPR25], which allows for strong adversaries.

Improved Certificates for Independence Number in Semirandom Hypergraphs

TL;DR

Two key improvements are made: firstly, sharper bounds are proved that get rid of pesky logarithmic factors in , and nearly attain the conjectured optimal computational threshold of , and secondly, robust Sum-of-Squares (SoS) certificates are designed, proving the authors' bounds in the more challenging semirandom hypergraph model.

Abstract

We study the problem of efficiently certifying upper bounds on the independence number of -uniform hypergraphs. This is a notoriously hard problem, with efficient algorithms failing to approximate the independence number within factor in the worst case [Has99, Zuc07]. We study the problem in random and semirandom hypergraphs. There is a folklore reduction to the graph case, achieving a certifiable bound of . More recently, the work [GKM22] improved this by constructing spectral certificates that yield a bound of . We make two key improvements: firstly, we prove sharper bounds that get rid of pesky logarithmic factors in , and nearly attain the conjectured optimal (in both and ) computational threshold of , and secondly, we design robust Sum-of-Squares (SoS) certificates, proving our bounds in the more challenging semirandom hypergraph model. Our analysis employs the proofs-to-algorithms paradigm [BS16, FKP19] in showing an upper bound for pseudo-expectation of degree- SoS relaxation of the natural polynomial system for maximum independent set. The challenging case is odd-arity hypergraphs, where we employ a tensor-based analysis that reduces the problem to proving bounds on a natural class of random chaos matrices associated with -uniform hypergraphs. Previous bounds [AMP21, RT23] have a logarithmic dependence, which we remove by leveraging recent progress on matrix concentration inequalities [BBvH23, BLNvH25]; we believe these may be useful in other hypergraph problems. As an application, we show our improved certificates can be combined with an SoS relaxation of a natural -coloring polynomial system to recover an arbitrary planted -colorable subhypergraph in a semirandom model along the lines of [LPR25], which allows for strong adversaries.
Paper Structure (54 sections, 35 theorems, 259 equations, 2 figures, 1 table)

This paper contains 54 sections, 35 theorems, 259 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Our Models and Results
  3. Motivation and Related Works
  4. Independent Sets in Hypergraphs.
  5. Certifying Independent Set in Random Hypergraphs.
  6. Recovery for Planted Problems.
  7. Semirandom Models
  8. Sum-of-Squares Hierarchy
  9. Sharper Matrix Norm Bounds.
  10. Technical Overview
  11. Analysis via High-Degree Moments.
  12. Analysis for Even Arity Case
  13. From all-distinct monomials to quadratic forms.
  14. SoS analysis reduces problem to spectral norm bounds.
  15. Bounding the spectral norm of Flattening Matrix
  16. ...and 39 more sections

Key Result

Theorem 4

For an instance of a random hypergraph $H=(V,E)$ generated from the semirandom hypergraph model $\mathcal{H}_1\left(n,\ell,p \right )$, there is an $n^{\mathcal{O}\left(\ell \right )}$ running time refutation algorithm such that w.h.p.,

Figures (2)

  • Figure 1: Landscape of refutation certificates (log scale): Even case ($\ell=6$) and Odd case ($\ell=7$).
  • Figure 2: Difficulties with Reduction from $2$-Colorable $4$-Uniform Planted Hypergraphs to Graphs

Theorems & Definitions (90)

  • Definition 1: Random $\ell$-Uniform Hypergraph
  • Definition 2: Semirandom Hypergraph Model
  • Definition 3: Refutation Certificate for Hypergraph Independence Number
  • Theorem 4
  • Remark 5
  • Remark 6
  • Remark 7
  • Corollary 8
  • Remark 9
  • Definition 10: Polynomial System
  • ...and 80 more