A Fast Maximum k-Plex Algorithm Parameterized by the Degeneracy Gap

Abstract

Given a graph, the k-plex is a vertex set 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 a given graph, is an important but computationally challenging problem in applications like graph search and community detection. So far, there is a number of empirical algorithms without sufficient theoretical explanations on the efficiency. We try to bridge this gap by defining a novel parameter of the input instance, g_k(G), the gap between the degeneracy bound and the size of maximum k-plex in the given graph, and presenting an exact algorithm parameterized by g_k(G). In other words, we design an algorithm with running time polynomial in the size of input graph and exponential in g_k(G) where k is a constant. Usually, g_k(G) is small and bounded by O(log(|V|)) in real-world graphs, indicating that the algorithm runs in polynomial time. We also carry out massive experiments and show that the algorithm is competitive with the state-of-the-art solvers. Additionally, for large k values such as 15 and 20, our algorithm has superior performance over existing algorithms.