Bi-Covering: Covering Edges with Two Small Subsets of Vertices

We study the following basic problem called Bi-Covering. Given a graph GpV,Eq, find two (not necessarily disjoint) sets A Ď V and B Ď V such that A Y B “ V and that every edge e belongs to either the graph induced by A or to the graph induced by B. The goal is to minimize maxt|A|, |B|u. This is the most simple case of the Channel Allocation problem [13]. A solution that outputs V,H gives ratio at most 2. We show that under the Strong Unique Game Conjecture by Bansal and Khot [6] there is no 2 ́ ratio algorithm for the problem, for any constant ą 0. Given a bipartite graph, Max-Bi-Clique is a problem of finding largest kˆk complete bipartite sub graph. For Max-Bi-Clique problem, a constant factor hardness was known under random 3-SAT hypothesis of Feige [10] and also under the assumption that NP Ę X ą0 BPTIMEp2 q [17]. It was an open problem in [3] to prove inapproximability of Max-Bi-Clique assuming weaker conjecture. Our result implies similar hardness result assuming the Strong Unique Games Conjecture. On the algorithmic side, we also give better than 2 approximation for Bi-Covering on numerous special graph classes. In particular, we get 1.876 approximation for Chordal graphs, exact algorithm for Interval Graphs, 1 ` op1q for Minor Free Graph, 2 ́ 4δ{3 for graphs with minimum degree δn, 2{p1 ` δ2{8q for δ-vertex expander, 8{5 for Split Graphs, 2 ́ p6{5q ̈ 1{d for graphs with minimum constant degree d etc. Our algorithmic results are quite non-trivial. In achieving these results, we use various known structural results about the graphs, combined with the techniques that we develop tailored to getting better than 2 approximation. 1998 ACM Subject Classification G.2.2 Graph Algorithms