Exact Clique Number Manipulation via Edge Interdiction

TL;DR

This paper introduces a novel exact algorithm for the Edge Interdiction Clique Problem, effectively reducing large graphs and solving the problem more efficiently than previous methods, with applications in network analysis.

Contribution

The paper develops new mixed integer linear formulations and a two-stage exact algorithm, RLCM, for solving the EICP more scalably and accurately than existing approaches.

Findings

RLCM outperforms existing methods on benchmark and real-world graphs.

The new formulations enable tighter modeling of clique-related inequalities.

Graph reduction techniques significantly improve computational efficiency.

Abstract

The Edge Interdiction Clique Problem (EICP) aims to remove at most $k$ edges from a graph so as to minimize the size of the largest clique in the remaining graph. This problem captures a fundamental question in graph manipulation: which edges are structurally critical for preserving large cliques? Such a problem is also motivated by practical applications including protein function maintenance and image matching. The EICP is computationally challenging and belongs to a complexity class beyond NP. Existing approaches rely on general mixed-integer bilevel programming solvers or reformulate the problem into a single-level mixed integer linear program. However, they are still not scalable when the graph size and interdiction budget $k$ grow. To overcome this, we investigate new mixed integer linear formulations, which recast the problem into a sequence of parameterized Edge Blocker Clique Problems (EBCP). This perspective decomposes the original problem into simpler subproblems and enables tighter modeling of clique-related inequalities. Furthermore, we propose a two-stage exact algorithm, \textsc{RLCM}, which first applies problem-specific reduction techniques to shrink the graph and then solves the reduced problem using a tailored branch-and-cut framework. Extensive computational experiments on maximum clique benchmark graphs, large real-world sparse networks, and random graphs demonstrate that \textsc{RLCM} consistently outperforms existing approaches.