Polynomial kernels for proper interval completion and related problems

Abstract

Given a graph G = (V,E) and a positive integer k, the Proper Interval Completion problem asks whether there exists a set F of at most k pairs of (V x V)\E such that the graph H = (V, E∪F) is a proper interval graph. The Proper Interval Completion problem finds applications in molecular biology and genomic research [11]. First announced by Kaplan, Tarjan and Shamir in FOCS '94, this problem is known to be FPT [11], but no polynomial kernel was known to exist. We settle this question by proving that Proper Interval Completion admits a kernel with O(k5) vertices. Moreover, we prove that a related problem, the so-called Bipartite Chain Deletion problem admits a kernel with O(k2) vertices, completing a previous result of Guo [10]. © 2011 Springer-Verlag.