Correlating Theory and Practice in Finding Clubs and Plexes

TL;DR

This paper explores the effectiveness of Turing kernelization in solving maximum s-clubs and s-plexes problems, providing new theoretical insights and practical evidence that better explain observed running times.

Contribution

It introduces a new parameterized analysis for Turing kernelization and a method to compare theoretical bounds with actual running times.

Findings

Turing kernelization improves practical solving of large subgraph problems.

New bounds better match observed running times than previous bounds.

Proposed method effectively compares theoretical and empirical performance.

Abstract

Finding large "cliquish" subgraphs is a classic NP-hard graph problem. In this work, we focus on finding maximum $s$-clubs and $s$-plexes, i.e., graphs of diameter $s$ and graphs where each vertex is adjacent to all but $s$ vertices. Preprocessing based on Turing kernelization is a standard tool to tackle these problems, especially on sparse graphs. We provide a new parameterized analysis for the Turing kernelization and demonstrate their usefulness in practice. Moreover, we provide evidence that the new theoretical bounds indeed better explain the observed running times than the existing theoretical running time bounds. To this end, we suggest a general method to compare how well theoretical running time bounds fit to measured running times.