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A Reduction-based Algorithm for the Clique Interdiction Problem

Chenghao Zhu, Yi Zhou, Haoyu Jiang

TL;DR

The paper addresses the Clique Interdiction Problem, a bilevel optimization task to minimize the maximum clique after removing up to $k$ vertices. It introduces RECIP, a reduction-based algorithm that uses six novel reduction rules to aggressively simplify graphs and tighten lower bounds via LP relaxations and flow-based techniques, integrating these into a branch-and-cut framework. Empirical results on 124 real-world networks show RECIP outperforms the state-of-the-art CLIQUE-INTER across most instances and budgets, often achieving substantial preprocessing (up to ~85% vertex reduction) and faster solution times. The work highlights both the practical power of data reductions in complex graph interdiction and the remaining challenges in dense graphs, proposing future directions for stronger reductions and enhanced integration with exact solvers. Overall, RECIP advances scalable CIP solving by combining principled reductions with rigorous optimization techniques and competitive empirical performance.

Abstract

The Clique Interdiction Problem (CIP) aims to minimize the size of the largest clique in a given graph by removing a given number of vertices. The CIP models a special Stackelberg game and has important applications in fields such as pandemic control and terrorist identification. However, the CIP is a bilevel graph optimization problem, making it very challenging to solve. Recently, data reduction techniques have been successfully applied in many (single-level) graph optimization problems like the vertex cover problem. Motivated by this, we investigate a set of novel reduction rules and design a reduction-based algorithm, RECIP, for practically solving the CIP. RECIP enjoys an effective preprocessing procedure that systematically reduces the input graph, making the problem much easier to solve. Extensive experiments on 124 large real-world networks demonstrate the superior performance of RECIP and validate the effectiveness of the proposed reduction rules.

A Reduction-based Algorithm for the Clique Interdiction Problem

TL;DR

The paper addresses the Clique Interdiction Problem, a bilevel optimization task to minimize the maximum clique after removing up to vertices. It introduces RECIP, a reduction-based algorithm that uses six novel reduction rules to aggressively simplify graphs and tighten lower bounds via LP relaxations and flow-based techniques, integrating these into a branch-and-cut framework. Empirical results on 124 real-world networks show RECIP outperforms the state-of-the-art CLIQUE-INTER across most instances and budgets, often achieving substantial preprocessing (up to ~85% vertex reduction) and faster solution times. The work highlights both the practical power of data reductions in complex graph interdiction and the remaining challenges in dense graphs, proposing future directions for stronger reductions and enhanced integration with exact solvers. Overall, RECIP advances scalable CIP solving by combining principled reductions with rigorous optimization techniques and competitive empirical performance.

Abstract

The Clique Interdiction Problem (CIP) aims to minimize the size of the largest clique in a given graph by removing a given number of vertices. The CIP models a special Stackelberg game and has important applications in fields such as pandemic control and terrorist identification. However, the CIP is a bilevel graph optimization problem, making it very challenging to solve. Recently, data reduction techniques have been successfully applied in many (single-level) graph optimization problems like the vertex cover problem. Motivated by this, we investigate a set of novel reduction rules and design a reduction-based algorithm, RECIP, for practically solving the CIP. RECIP enjoys an effective preprocessing procedure that systematically reduces the input graph, making the problem much easier to solve. Extensive experiments on 124 large real-world networks demonstrate the superior performance of RECIP and validate the effectiveness of the proposed reduction rules.
Paper Structure (35 sections, 15 theorems, 2 equations, 6 figures, 2 tables, 3 algorithms)

This paper contains 35 sections, 15 theorems, 2 equations, 6 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Problem Background
  3. Pandemic Control
  4. Terrorist Identification
  5. Related Literature
  6. Our Contributions
  7. Preliminary
  8. Reduction Rules
  9. Exact Clique Reduction
  10. Degree Reduction
  11. Color Reduction
  12. Triangle Reduction
  13. Interdiction Reduction
  14. Domination Reduction
  15. The Reduction-and-Search Algorithm
  16. ...and 20 more sections

Key Result

Lemma 1

Given a graph $G=(V,E)$, an integer $k$, and an integer $lb$, if $\theta(G,k)\ge lb$ and there is a vertex $u\in V$ such that $\omega(G[N(u)]) \leq lb-2$, then $\theta(G,k)=\theta(G[V\setminus \{u\}])$.

Figures (6)

  • Figure 1: Structure of the network flow model, the value on the side represents the flow size.
  • Figure 2: Flowchart of the reduction-based preprocess, time complexities are given to each step.
  • Figure 3: The proportion of instances solved by both algorithms within the time range of 0.1 to 600 seconds on real-world network graphs under different $k$ values.
  • Figure 4: Runtime of both algorithms for each instance, with instances exceeding the 600-second time limit plotted at the boundary. For clarity, a dashed line representing $x = y$ is added.
  • Figure 5: Reduction proportion at each step for 10 selected graphs and the average, with different $k$ values.
  • ...and 1 more figures

Theorems & Definitions (20)

  • Lemma 1: Exact Clique Reduction
  • Lemma 2
  • Lemma 3: Degree Reduction
  • Lemma 4: Color Reduction
  • Lemma 5: Triangle Reduction
  • Lemma 6: Triangle Clique Reduction
  • Lemma 7: Triangle Color Reduction
  • Lemma 8: Interdiction Reduction
  • Lemma 9: Domination Reduction
  • Lemma 10
  • ...and 10 more