Exact Clique Number Manipulation via Edge Interdiction

2026-01-05

arXiv.org

AI Summary

This study addresses the edge-deletion clique problem: minimizing the size of the maximum clique in a graph by removing at most \(k\) edges, thereby identifying critical edges that sustain large clique structures. To this end, the authors propose a novel mixed-integer linear programming (MILP) formulation enhanced with graph reduction techniques and a two-phase exact algorithm named RLCM. The approach integrates clique inequalities to strengthen the branch-and-cut procedure. Extensive experiments demonstrate that the method substantially outperforms existing algorithms across standard maximum clique benchmarks, real-world sparse networks, and random graphs, achieving significant improvements in both computational efficiency and scalability.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Graph-based Machine Learning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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.

Problem

• Edge Interdiction
• Clique Number
• Graph Manipulation
• Combinatorial Optimization
• NP-hard

Innovation

• Edge Interdiction
• Clique Number Minimization
• Mixed Integer Linear Programming
• Branch-and-Cut
• Graph Reduction