Nearly optimal NP-hardness of vertex cover on k-uniform k-partite hypergraphs

• Authors: Sushant Sachdeva;Rishi Saket
• Venue: APPROX'11/RANDOM'11 Proceedings of the 14th international workshop and 15th international conference on Approximation, randomization, and combinatorial optimization: algorithms and techniques
• Year: 2011

We study the problem of computing the minimum vertex cover on kuniform 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 k/2 -approximation based on the standard LP relaxation. Subsequent work by Aharoni, Holzman and Krivelevich [1] 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 nontrivial NP-hardness of approximation factor of k/4-ε was shown in a recent work by Guruswami and Saket [13]. They also showed that assuming Khot's Unique Games Conjecture yields a k/2- ε 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 k/2 - 1 + 1/2k - ε. 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.