Nearly Optimal NP-Hardness of Vertex Cover on k-Uniform k-Partite Hypergraphs

TL;DR

This paper establishes nearly tight NP-hardness bounds for approximating the minimum vertex cover on k-uniform k-partite hypergraphs, improving understanding of the problem's computational complexity.

Contribution

It proves that approximating the vertex cover within a factor of k/2 - 1 + 1/(2k) - ε is NP-hard, nearly matching known approximation algorithms.

Findings

Hardness factor is off from optimal by at most 1 for k ≥ 4.

Uses Multi-Layered PCP and biased Long Codes in the reduction.

Analyzes cross-intersecting set families in extremal set theory.

Abstract

We study the problem of computing the minimum vertex cover on k-uniform k-partite hypergraphs when the k-partition is given. On bipartite graphs (k = 2), the minimum vertex cover can be computed in polynomial time. For general k, the problem was studied by Lov\'asz, who gave a k/2 -approximation based on the standard LP relaxation. Subsequent work by Aharoni, Holzman and Krivelevich showed a tight integrality gap of (k/2 - o(1)) for the LP relaxation. While this problem was known to be NP-hard for k >= 3, the first non-trivial NP-hardness of approximation factor of k/4- \eps was shown in a recent work by Guruswami and Saket. They also showed that assuming Khot's Unique Games Conjecture yields a k/2 - \eps inapproximability for this problem, implying the optimality of Lov\'asz's result. In this work, we show that this problem is NP-hard to approximate within k/2- 1 + 1/2k -\eps. This hardness factor is off from the optimal by an additive constant of at most 1 for k >= 4. Our reduction relies on the Multi-Layered PCP of Dinur et al. and uses a gadget - based on biased Long Codes - adapted from the LP integrality gap of Aharoni et al. The nature of our reduction requires the analysis of several Long Codes with different biases, for which we prove structural properties of the so called cross-intersecting collections of set families - variants of which have been studied in extremal set theory.