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# Constrained Bipartite Vertex Cover: The Easy Kernel is Essentially Tight

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## Cite As

Bart M. P. Jansen. Constrained Bipartite Vertex Cover: The Easy Kernel is Essentially Tight. In 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 47, pp. 45:1-45:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
https://doi.org/10.4230/LIPIcs.STACS.2016.45

## Abstract

The CONSTRAINED BIPARTITE VERTEX COVER problem asks, for a bipartite graph G with partite sets A and B, and integers k_A and k_B, whether there is a vertex cover for G containing at most k_A vertices from A and k_B vertices from B. The problem has an easy kernel with 2 * k_A * k_B edges and 4 k_A * k_B vertices, based on the fact that every vertex in A of degree more than k_B has to be included in the solution, together with every vertex in B of degree more than k_A. We show that the number of vertices and edges in this kernel are asymptotically essentially optimal in terms of the product k_A * k_B. We prove that if there is a polynomial-time algorithm that reduces any instance (G,A,B,k_A,k_B) of CONSTRAINED BIPARTITE VERTEX COVER to an equivalent instance (G',A',B',k'_A,k'_B) such that k'_A in (k_A)^{O(1)}, k'_B in (k_B)^{O(1)}, and |V(G')| in O((k_A * k_B)^{1 - epsilon}), for some epsilon > 0, then NP subseteq 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(n^{2- epsilon}) bits, then NP subseteq coNP/poly.
##### Keywords
• kernel lower bounds
• constrained bipartite vertex cover

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