Polynomial kernels for Proper Interval Completion and related problems

TL;DR

This paper proves that the Proper Interval Completion problem has a polynomial kernel of size O(k^5), and also establishes a kernel of size O(k^2) for the related Bipartite Chain Deletion problem, advancing kernelization theory.

Contribution

The paper provides the first polynomial kernels for Proper Interval Completion and improves kernel bounds for Bipartite Chain Deletion, resolving open questions in parameterized complexity.

Findings

Proper Interval Completion admits an O(k^5) vertex kernel.

Bipartite Chain Deletion admits an O(k^2) vertex kernel.

These results settle longstanding open problems in kernelization.

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 \times V)\E such that the graph H = (V,E \cup F) is a proper interval graph. The Proper Interval Completion problem finds applications in molecular biology and genomic research. First announced by Kaplan, Tarjan and Shamir in FOCS '94, this problem is known to be FPT, but no polynomial kernel was known to exist. We settle this question by proving that Proper Interval Completion admits a kernel with at most O(k^5) vertices. Moreover, we prove that a related problem, the so-called Bipartite Chain Deletion problem, admits a kernel with at most O(k^2) vertices, completing a previous result of Guo.