Independent Sets in Semi-random Hypergraphs

TL;DR

This paper extends algorithms for finding large independent sets from semi-random graphs to hypergraphs, using advanced SDP and hierarchy techniques to improve recovery in semi-random models.

Contribution

It generalizes existing semi-random graph algorithms to hypergraphs, incorporating Lasserre/SoS hierarchies for better independent set approximation.

Findings

Algorithms successfully recover large independent sets in hypergraphs.

Extension of semi-random models from graphs to hypergraphs.

Use of advanced SDP hierarchies enhances approximation guarantees.

Abstract

A set of vertices in a hypergraph is called an independent set if no hyperedge is completely contained inside the set. Given a hypergraph, computing its largest size independent set is an NP-hard problem. In this work, we study the independent set problem on hypergraphs in a natural semi-random family of instances. Our semi-random model is inspired by the Feige-Kilian model [FK01]. This popular model has also been studied in the works of [FK01, Ste17, MMT20] etc. McKenzie, Mehta, and Trevisan [MMT20] gave algorithms for computing independent sets in such a semi-random family of graphs. The algorithms by McKenzie et al. [MMT20] are based on rounding a "crude-SDP". We generalize their results and techniques to hypergraphs for an analogous family of hypergraph instances. Our algorithms are based on rounding the "crude-SDP" of McKenzie et al. [MMT20], augmented with "Lasserre/SoS like" hierarchy of constraints. Analogous to the results of McKenzie et al. [MMT20], we study the ranges of input parameters where we can recover the planted independent set or a large independent set.