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 polynomialtime 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 polynomialtime hierarchy collapses. Using a different construction, we prove that if there is a polynomialtime algorithm that reduces any nvertex 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.
BibTeX  Entry
@InProceedings{jansen:LIPIcs:2016:5746,
author = {Bart M. P. Jansen},
title = {{Constrained Bipartite Vertex Cover: The Easy Kernel is Essentially Tight}},
booktitle = {33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)},
pages = {45:145:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770019},
ISSN = {18688969},
year = {2016},
volume = {47},
editor = {Nicolas Ollinger and Heribert Vollmer},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2016/5746},
URN = {urn:nbn:de:0030drops57463},
doi = {10.4230/LIPIcs.STACS.2016.45},
annote = {Keywords: kernel lower bounds, constrained bipartite vertex cover}
}
Keywords: 

kernel lower bounds, constrained bipartite vertex cover 
Collection: 

33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016) 
Issue Date: 

2016 
Date of publication: 

16.02.2016 