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

Abstract

Abstract. Computing a minimum vertex cover in graphs and hypergraphs is a well-studied optimizaton problem. While intractable in general, it is well known that on bipartite graphs, vertex cover is polynomial time solvable. In this work, we study the natural extension of bipartite vertex cover to hypergraphs, namely finding a small vertex cover in kuniform k-partite hypergraphs, when the k-partition is given as input. For this problem Lovász [16] gave a k 2 factor LP rounding based approximation, and a matching k 2 − o(1) integrality gap instance was constructed by Aharoni et al. [1]. We prove the following results, which are the first strong hardness results for this problem (here ε > 0 is an arbitrary constant):-NP-hardness of approximating within a factor of k 4 − ε , and -Unique Games-hardness of approximating within a factor of k 2 − ε , showing optimality of Lovász's algorithm under the Unique Games conjecture. 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]. The modification is to ensure that the reduction preserves the desired structural properties of the hypergraph.

On LP-based Approximability for Strict CSPs

“…In particular, we show that a natural linear program (LP) captures precisely (up-to arbitrarily small additive error) the approximability of strict-CSPs such as covering-packing problems, which include Vertex Cover, Hypergraph Vertex Cover and Independent Set, as observed by Guruswami and Saket [GS10] -the k-partite-k-uniform-Hypergraph Vertex Cover problem, and the concurrent open shop problem in scheduling [MQS + 09], [BK09a]. We show how to convert integrality gap for the LP for these problems to a Unique Games-based hardness of approximation result in a principled way.…”

“…The advantage of our approach is that it converts any integrality gap into an inapproximability result. Moreover, since the reduction inherits the structure of the integrality-gap, our result has been used to derive new optimal inapproximability result for the kpartite-k-uniform-Hypergraph Vertex Cover problem by Guruswami and Saket [GS10] discussed later in this section.…”

“…A nice feature about composing integrality gaps with Unique Games-instances is that some structure of the integrality gap shows up in the final instance. Guruswami and Saket [GS10] considered the problem of k-partite-k-uniform-Hypergraph Vertex Cover, where, in addition to the vertices and the hyper-edges, one is also given a k-partition of the vertex set and each hyper-edge contains exactly one vertex from a partition. As proved by Lovasz [Lov75], this problem has a k/2-approximation algorithm.…”

Ant Colony Optimization and hypergraph covering problems

“…Independent set problems are also related to covering problems. The decision versions of these problems are NP-complete, while the optimization versions (minimum vertex/edge cover/maximum independent set) are NP-Hard [29]. The vertex cover problem is also known as transversal or hitting set problem, where the edge cover problem is a special case of set cover problem for hypergraphs.…”

“…Okun [25] discusses the approximation of hypergraph vertex cover for hypergraphs of bounded degree and bounded number of neighbouring vertices. Complexity and approximation results for the connected vertex cover problem on graphs and hypergraphs [30], vertex cover on dense hypergraphs [27], set covering on hypergraphs [28] and vertex cover on k-partite k-uniform hypergraphs [29] have been obtained. Independent set problems are closely related to covering problems.…”