Fast Maximum $k$-Plex Algorithms Parameterized by Small Degeneracy Gaps
This addresses a computationally challenging problem in graph mining and community detection, offering theoretical explanations for efficiency, but it is incremental as it builds on existing practical algorithms with new parameters.
The paper tackles the maximum k-plex problem in graphs by introducing a novel parameter, the degeneracy gap, and presents an exact algorithm with worst-case running time polynomial in graph size and exponential in this gap, which is small in real-world inputs, leading to competitive performance, especially dominating existing algorithms for large k values like 15 and 20.
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.