Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Low-Degree Hardness of Finding Large Independent Sets in Sparse Random Hypergraphs

Abhishek Dhawan, Yuzhou Wang

Abstract

We study the algorithmic task of finding large independent sets in Erdos-Renyi $r$-uniform hypergraphs on $n$ vertices having average degree $d$. Krivelevich and Sudakov showed that the maximum independent set has density $\left(\frac{r\log d}{(r-1)d}\right)^{1/(r-1)}$. We show that the class of low-degree polynomial algorithms can find independent sets of density $\left(\frac{\log d}{(r-1)d}\right)^{1/(r-1)}$ but no larger. This extends and generalizes earlier results of Gamarnik and Sudan, Rahman and Virag, and Wein on graphs, and answers a question of Bal and Bennett. We conjecture that this statistical-computational gap holds for this problem. Additionally, we explore the universality of this gap by examining $r$-partite hypergraphs. A hypergraph $H=(V,E)$ is $r$-partite if there is a partition $V=V_1\cup\cdots\cup V_r$ such that each edge contains exactly one vertex from each set $V_i$. We consider the problem of finding large balanced independent sets (independent sets containing the same number of vertices in each partition) in random $r$-partite hypergraphs with $n$ vertices in each partition and average degree $d$. We prove that the maximum balanced independent set has density $\left(\frac{r\log d}{(r-1)d}\right)^{1/(r-1)}$ asymptotically. Furthermore, we prove an analogous low-degree computational threshold of $\left(\frac{\log d}{(r-1)d}\right)^{1/(r-1)}$. Our results recover and generalize recent work of Perkins and the second author on bipartite graphs. While the graph case has been extensively studied, this work is the first to consider statistical-computational gaps of optimization problems on random hypergraphs. Our results suggest that these gaps persist for larger uniformities as well as across many models. A somewhat surprising aspect of the gap for balanced independent sets is that the algorithm achieving the lower bound is a simple degree-1 polynomial.

The Low-Degree Hardness of Finding Large Independent Sets in Sparse Random Hypergraphs

Abstract

We study the algorithmic task of finding large independent sets in Erdos-Renyi -uniform hypergraphs on vertices having average degree . Krivelevich and Sudakov showed that the maximum independent set has density . We show that the class of low-degree polynomial algorithms can find independent sets of density but no larger. This extends and generalizes earlier results of Gamarnik and Sudan, Rahman and Virag, and Wein on graphs, and answers a question of Bal and Bennett. We conjecture that this statistical-computational gap holds for this problem. Additionally, we explore the universality of this gap by examining -partite hypergraphs. A hypergraph is -partite if there is a partition such that each edge contains exactly one vertex from each set . We consider the problem of finding large balanced independent sets (independent sets containing the same number of vertices in each partition) in random -partite hypergraphs with vertices in each partition and average degree . We prove that the maximum balanced independent set has density asymptotically. Furthermore, we prove an analogous low-degree computational threshold of . Our results recover and generalize recent work of Perkins and the second author on bipartite graphs. While the graph case has been extensively studied, this work is the first to consider statistical-computational gaps of optimization problems on random hypergraphs. Our results suggest that these gaps persist for larger uniformities as well as across many models. A somewhat surprising aspect of the gap for balanced independent sets is that the algorithm achieving the lower bound is a simple degree-1 polynomial.
Paper Structure (27 sections, 29 theorems, 154 equations)

This paper contains 27 sections, 29 theorems, 154 equations.

Table of Contents

  1. Introduction
  2. Relation to Prior Work
  3. Statistical-computational gaps.
  4. Low-degree algorithms.
  5. Statistical inference on random hypergraphs.
  6. The Low-Degree Framework
  7. Main Results
  8. Proof Overview
  9. Independent Sets in $\mathcal{H}_{r}(n,p)$.
  10. Balanced Independent sets in $\mathcal{H}(r,n,p)$.
  11. Concluding Remarks
  12. Preliminaries
  13. Notation
  14. Local Algorithms on Hypergraphs
  15. Interpolating Paths and Stable Algorithms
  16. ...and 12 more sections

Key Result

Theorem 1

Let $\varepsilon > 0$ and let $r \geq 2$. In the double limit $n \to \infty$ followed by $d \to \infty$, the following hold for $p = d/\binom{n-1}{r-1}$.

Theorems & Definitions (65)

  • Theorem : Informal version of Theorems \ref{['theorem: low-deg hypergraph achievability']} and \ref{['theorem: low-deg hypergraph impossibility']}
  • Conjecture 1
  • Theorem : Informal version of Theorems \ref{['theorem: stat thresh balanced']} and \ref{['theorem: low-deg thresh balanced']}
  • Conjecture 2
  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4: The Random Hypergraph Models
  • Theorem 1.5: Achievability result for $\mathcal{H}_{r}(n,p)$
  • Theorem 1.6: Impossibility result for $\mathcal{H}_{r}(n,p)$
  • ...and 55 more