Hardness of Finding Independent Sets in 2-Colorable and Almost 2-Colorable Hypergraphs

TL;DR

This paper establishes new hardness results for finding large independent sets in hypergraphs that are nearly 2-colorable, advancing understanding of computational difficulty in hypergraph coloring problems.

Contribution

It proves the first unconditional hardness results for independent sets in almost 2-colorable 3-uniform hypergraphs and introduces nearly polynomial hardness bounds for 4-uniform hypergraphs, bridging previous gaps.

Findings

Hardness of finding large independent sets in almost 2-colorable 4-uniform hypergraphs.

NP-hardness of finding large independent sets in almost 2-colorable 3-uniform hypergraphs.

Conditional hardness results assuming the d-to-1 Games Conjecture for 2-colorable 3-uniform hypergraphs.

Abstract

This work studies the hardness of finding independent sets in hypergraphs which are either 2-colorable or are almost 2-colorable, i.e. can be 2-colored after removing a small fraction of vertices and the incident hyperedges. To be precise, say that a hypergraph is (1-eps)-almost 2-colorable if removing an eps fraction of its vertices and all hyperedges incident on them makes the remaining hypergraph 2-colorable. In particular we prove the following results. For an arbitrarily small constant gamma > 0, there is a constant xi > 0, such that, given a 4-uniform hypergraph on n vertices which is (1 - eps)-almost 2-colorable for eps = 2^{-(log n)^xi}, it is quasi-NP-hard to find an independent set of n/(2^{(log n)^{1-gamma}}) vertices. For any constants eps, delta > 0, given as input a 3-uniform hypergraph on $n$ vertices which is (1-eps)-almost 2-colorable, it is NP-hard to find an independent set of delta n vertices. Assuming the d-to-1 Games Conjecture the following holds. For any constant delta > 0, given a 2-colorable 3-uniform hypergraph on n vertices, it is NP-hard to find an independent set of delta n vertices. The hardness result on independent set in almost 2-colorable 3-uniform hypergraphs was earlier known only assuming the Unique Games Conjecture. In this work we prove the result unconditionally. For independent sets in 2-colorable 3-uniform hypergaphs we prove the first strong hardness result, albeit assuming the d-to-1 Games Conjecture. Our result on almost 2-colorable 4-uniform hypergraphs gives the first nearly polynomial hardness factor for independent set in hypergraphs which are (almost) colorable with constantly many colors. It partially bridges the gap between the previous best lower bound of poly(log n) and the algorithmic upper bounds of n^{Omega(1)}.