Win-Win Kernelization for Degree Sequence Completion Problems

TL;DR

This paper introduces a kernelization approach for degree sequence completion problems, providing fixed-parameter algorithms for edge addition cases and establishing polynomial kernelizability, thus advancing preprocessing techniques for NP-hard graph problems.

Contribution

The paper presents a novel kernelization method for degree sequence completion problems, transforming graph problems into number problems and solving them efficiently.

Findings

Fixed-parameter tractability for edge addition problems.

No similar results for vertex or edge deletion cases.

Polynomial kernelization for Degree Constraint Edge Addition.

Abstract

We study provably effective and efficient data reduction for a class of NP-hard graph modification problems based on vertex degree properties. We show fixed-parameter tractability for NP-hard graph completion (that is, edge addition) cases while we show that there is no hope to achieve analogous results for the corresponding vertex or edge deletion versions. Our algorithms are based on transforming graph completion problems into efficiently solvable number problems and exploiting f-factor computations for translating the results back into the graph setting. Our core observation is that we encounter a win-win situation: either the number of edge additions is small or the problem is polynomial-time solvable. This approach helps in answering an open question by Mathieson and Szeider [JCSS 2012] concerning the polynomial kernelizability of Degree Constraint Edge Addition and leads to a general method of approaching polynomial-time preprocessing for a wider class of degree sequence completion problems.