Fast Maximum $k$-Plex Algorithms Parameterized by Small Degeneracy Gaps

TL;DR

This paper introduces a new parameterized algorithm for the maximum $k$-plex problem that is efficient on real-world graphs with small degeneracy gaps, outperforming existing methods especially for large $k$ values.

Contribution

The paper proposes a novel parameter, $g_k(G)$, and an exact algorithm for the maximum $k$-plex problem that is efficient when this parameter is small, supported by extensive empirical evaluation.

Findings

Algorithm runs in polynomial time for real-world graphs with small $g_k(G)$

Outperforms state-of-the-art algorithms for large $k$ values like 15 and 20

Empirical analysis confirms the effectiveness of the parameters and implementation components.

Abstract

Given a graph, a $k$-plex is a set of vertices in which each vertex is not adjacent to at most $k-1$ other vertices in the set. The maximum $k$-plex problem, which asks for the largest $k$-plex from the given graph, is an important but computationally challenging problem in applications such as graph mining and community detection. So far, there are many practical algorithms, but without providing theoretical explanations on their efficiency. We define a novel parameter of the input instance, $g_k(G)$, the gap between the degeneracy bound and the size of the maximum $k$-plex in the given graph, and present an exact algorithm parameterized by this $g_k(G)$, which has a worst-case running time polynomial in the size of the input graph and exponential in $g_k(G)$. In real-world inputs, $g_k(G)$ is very small, usually bounded by $O(\log{(|V|)})$, indicating that the algorithm runs in polynomial time. We further extend our discussion to an even smaller parameter $cg_k(G)$, the gap between the community-degeneracy bound and the size of the maximum $k$-plex, and show that without much modification, our algorithm can also be parameterized by $cg_k(G)$. To verify the empirical performance of these algorithms, we carry out extensive experiments to show that these algorithms are competitive with the state-of-the-art algorithms. In particular, for large $k$ values such as $15$ and $20$, our algorithms dominate the existing algorithms. Finally, empirical analysis is performed to illustrate the effectiveness of the parameters and other key components in the implementation.