Constrained Bipartite Vertex Cover: The Easy Kernel is Essentially Tight

The Constrained Bipartite Vertex Cover problem asks, for a bipartite graph G with partite sets A and B, and integers kA and kB , whether there is a vertex cover for G containing at most kA vertices from A and kB vertices from B. The problem has an easy kernel with 2kA · kB edges and 4kA · kB vertices, based on the fact that every vertex in A of degree more than kB has to be included in the solution, together with every vertex in B of degree more than kA. We show that the number of vertices and edges in this kernel are asymptotically essentially optimal in terms of the product kA · kB . We prove that if there is a polynomial-time algorithm that reduces any instance (G,A,B, kA, kB) of Constrained Bipartite Vertex Cover to an equivalent instance (G′, A′, B′, k′ A, k′ B) such that k′ A ∈ (kA), k′ B ∈ (kB), and |V (G′)| ∈ O((kA · kB)), for some ε > 0, then NP ⊆ coNP/poly and the polynomial-time hierarchy collapses. Using a different construction, we prove that if there is a polynomial-time algorithm that reduces any n-vertex instance into an equivalent instance (of a possibly different problem) that can be encoded in O(n2−ε) bits, then NP ⊆ coNP/poly. 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems, G.2.2 Graph