Date of Original Version
72010
Type
Conference Proceeding
Rights Management
The final publication is available at Springer via http://dx.doi.org/10.1007/9783642141652_31
Abstract or Description
Computing a minimum vertex cover in graphs and hypergraphs is a wellstudied 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 kpartite hypergraphs, when the kpartition is given as input. For this problem Lovász [16] gave a k2 factor LP rounding based approximation, and a matching (k2−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):

NPhardness of approximating within a factor of (k4−ε), and

Unique Gameshardness of approximating within a factor of (k2−ε), showing optimality of Lovász’s algorithm under the Unique Games conjecture.
The NPhardness result is based on a reduction from minimum vertex cover in runiform hypergraphs for which NPhardness of approximating within r–1–ε was shown by Dinur et al.[5]. The Unique Gameshardness 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.
DOI
10.1007/9783642141652_31
Published In
Lecture Notes in Computer Science, 6198, 360371.