Inapproximability of Minimum Vertex Cover on k-Uniform k-Partite Hypergraphs

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 k ≥ 3, this problem is known to be NP-hard. For general k, the problem was studied by Lovász [23], who gave a k2 -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. We further investigate the inapproximability of minimum vertex cover on k-uniform k-partite hypergraphs and present the following results (here ε > 0 is an arbitrarily small constant): • NP-hardness of obtaining an approximation factor of ( k 4 − ε ) for even k, and ( k 4 − 1 4k − ε ) for odd k, • NP-hardness of obtaining a nearly-optimal approximation factor of ( k 2 − 1 + 1 2k − ε ) , and, • An optimal Unique Games-hardness for approximation within factor ( k 2 − ε ) , showing the optimality of Lovász’s algorithm if one assumes the Unique Games conjecture. The first 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. [8]. We include it for its simplicity, despite it being subsumed by the second hardness result. The Unique Games-hardness result is obtained by applying the results of Kumar et al. [22], with a slight modification, to the LP integrality gap due to Aharoni et al. [1]. The modification ensures that the reduction preserves the desired structural properties of the hypergraph. The reduction for the nearly optimal NP-hardness result relies on the Multi-Layered PCP of [8], and uses a gadget based on biased Long Codes, which is 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. ∗A preliminary version of these results appear in [13, 27]. †Computer Science Department, Carnegie Mellon University, PA, USA. [email protected]. Research supported in part by NSF CCF-1115525. ‡Department of Computer Science, Princeton University, NJ, USA. [email protected]. Research supported by NSF grants CCF-0832797 and CCF-1117309, and Sanjeev Arora’s Simons Investigator Grant. §IBM Thomas J. Watson Research Center, NY, USA. [email protected] ISSN 1433-8092 Electronic Colloquium on Computational Complexity, Report No. 71 (2013)