Bicovering: Covering edges with two small subsets of vertices

We study the following basic problem called Bi-Covering. Given a graph G(V,E), find two (not necessarily disjoint) sets A ⊆ V and B ⊆ V such that A ∪ 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 max{|A|, |B|}. This is the most simple case of the Channel Allocation problem [Gandhi et. al, Networks, 2006]. A solution that outputs V, 0 gives ratio at most 2. We show that under the similar Strong Unique Game Conjecture by [Bansal-Khot, FOCS, 2009] there is no 2 - ϵ ratio algorithm for the problem, for any constant ϵ > 0. Given a bipartite graph, Max-bi-clique is a problem of finding largest k × k complete bipartite sub graph. For Max-bi-clique problem, a constant factor hardness was known under random 3-SAT hypothesis of Feige [Feige, STOC, 2002] and also under the assumption that NP ⊈ ∩ϵ>0 BPTIME(2^nϵ) [Khot, SIAM J. on Comp., 2011]. It was an open problem in [Ambühl et. al., SIAM J. on Comp., 2011] 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+o(1) for Minor Free Graph, 2-4δ/3 for graphs with minimum degree δn, 2/(1+δ²/8) for δ-vertex expander, 8/5 for Split Graphs, 2-(6/5) · 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.