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

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 Lovász [23], who gave a $\frac{k}{2}$-approximation based on the standard LP relaxation. Subsequent work by Aharoni, Holzman and Krivelevich [1] showed a tight integrality gap of $\left(\frac{k}{2} - o(1)\right)$ 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 $\frac{k}{4}-\varepsilon$ was shown in a recent work by Guruswami and Saket [13]. They also showed that assuming Khot’s Unique Games Conjecture yields a $\frac{k}{2}-\varepsilon$ inapproximability for this problem, implying the optimality of Lovász’s result.

In this work, we show that this problem is NP-hard to approximate within $\frac{k}{2}-1+\frac{1}{2k}-\varepsilon$. 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 [8] and uses a gadget – based on biased Long Codes – adapted from the LP integrality gap of [1]. 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.

Research supported in part by NSF grants CCF-0832797, 0830673, and 0528414.