An Implicit Enumeration Approach for Maximum Ratio Clique Relaxations

ABSTRACT This article proposes an implicit enumeration approach to solve the maximum ratio ‐plex and the maximum ratio ‐defective clique problems. The approach is inspired by the classical Bron‐Kerbosch algorithm for enumerating all maximal cliques in a graph, which is extended to enumerating structures that are hereditary on induced subgraphs. Such structures include ‐plexes and ‐defective cliques, among many others. The performance of the proposed approach is compared with that of the methods based on mixed integer linear programming (MILP), binary search, and Newton's iteration through numerical experiments on randomly generated and real‐life network instances.