A New Multilayered PCP and the Hardness of Hypergraph Vertex Cover

Abstract

Given a k-uniform hyper-graph, the Ek-Vertex-Cover problem is to find the smallest subset of vertices that intersects every hyper-edge. We present a new multilayered PCP construction that extends the Raz verifier. This enables us to prove that Ek-Vertex-Cover is NP-hard to approximate within factor (k − 1 − ε) for any k ≥ 3 and any ε > 0. The result is essentially tight as this problem can be easily approximated within factor k. Our construction makes use of the biased Long-Code and is analyzed using combinatorial properties of s-wise t-intersecting families of subsets.

On the Inapproximability of Vertex Cover on k-Partite k-Uniform Hypergraphs

“…(of Theorem 7) Theorem 11 combined with the k−1−ε inapproximability for HypVC(k) given by [5] and stated in Theorem 5, implies an inapproximability of,…”

“…For general k, an Ω(k 1/19 ) hardness factor was first shown by Trevisan [21], subsequently strengthened to Ω(k 1−ε ) by Holmerin [10] and to a k − 3 − ε hardness factor due to Dinur, Guruswami and Khot [4]. The currently best known k − 1 − ε hardness factor is due to Dinur, Guruswami, Khot and Regev [5] who build upon [4] and the seminal work of Dinur and Safra [6] who showed the best known 1.36 hardness of approximation for vertex cover in graphs (k = 2).…”

“…The result for k-uniform k-partite hypergraphs follows by a simple reduction. Our results are based on a reduction from minimum vertex cover in k-uniform hypergraphs for which, as mentioned above, the best known factor k − 1 − ε hardness of approximation factor was given in [5].…”

“…The NP-hardness result is based on a reduction from minimum vertex cover in r-uniform hypergraphs for which NP-hardness of approximating within r−1−ε was shown by Dinur et al [5]. The Unique Games-hardness result is obtained by applying the recent results of Kumar et al [15], with a slight modification, to the LP integrality gap due to Aharoni et al [1].…”

Inapproximability of Hypergraph Vertex Cover and Applications to Scheduling Problems

“…On the inapproximability side, for the case k = 2, Dinur and Safra [5] obtained 1.36 NP-hardness improving on the 7 6 − ε hardness by Håstad [10]. For larger k, a sequence of works obtained hardness factor k 1/19 (Trevisan [22]), 2 − ε for some constant k (Goldreich [8]), 2 − ε for k = 4, 3 2 − ε for k = 3, k 1−o(1) (Holmerin [12,11]), k − 3 − ε (Dinur et al [3]), and finally k − 1 − ε (Dinur et al [4]) that superseded all earlier results for k ≥ 3.…”