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KD-Club: An Efficient Exact Algorithm with New Coloring-based Upper Bound for the Maximum k-Defective Clique Problem

Mingming Jin, Jiongzhi Zheng, Kun He

TL;DR

This work tackles the NP-hard Maximum $k$-Defective Clique Problem (MDCP) by introducing CLUB, a coloring-based upper bound that captures missing edges not only between the growing clique and the remaining candidates but also among candidates themselves. Building on CLUB, the authors propose KD-Club, an exact branch-and-bound algorithm that leverages CLUB both in preprocessing and during search to prune dramatically, achieving tighter bounds and smaller search trees. Extensive experiments on diverse, large-scale benchmarks demonstrate that KD-Club solves significantly more instances and usually with shorter running times than state-of-the-art solvers MADEC$^+$ and KDBB, especially for larger $k$ and dense graphs. The approach offers robust performance across sparse and dense graphs and provides a framework that could be adapted to tighten upper bounds in other relaxation-based combinatorial problems.

Abstract

The Maximum k-Defective Clique Problem (MDCP) aims to find a maximum k-defective clique in a given graph, where a k-defective clique is a relaxation clique missing at most k edges. MDCP is NP-hard and finds many real-world applications in analyzing dense but not necessarily complete subgraphs. Exact algorithms for MDCP mainly follow the Branch-and-bound (BnB) framework, whose performance heavily depends on the quality of the upper bound on the cardinality of a maximum k-defective clique. The state-of-the-art BnB MDCP algorithms calculate the upper bound quickly but conservatively as they ignore many possible missing edges. In this paper, we propose a novel CoLoring-based Upper Bound (CLUB) that uses graph coloring techniques to detect independent sets so as to detect missing edges ignored by the previous methods. We then develop a new BnB algorithm for MDCP, called KD-Club, using CLUB in both the preprocessing stage for graph reduction and the BnB searching process for branch pruning. Extensive experiments show that KD-Club significantly outperforms state-of-the-art BnB MDCP algorithms on the number of solved instances within the cut-off time, having much smaller search tree and shorter solving time on various benchmarks.

KD-Club: An Efficient Exact Algorithm with New Coloring-based Upper Bound for the Maximum k-Defective Clique Problem

TL;DR

This work tackles the NP-hard Maximum -Defective Clique Problem (MDCP) by introducing CLUB, a coloring-based upper bound that captures missing edges not only between the growing clique and the remaining candidates but also among candidates themselves. Building on CLUB, the authors propose KD-Club, an exact branch-and-bound algorithm that leverages CLUB both in preprocessing and during search to prune dramatically, achieving tighter bounds and smaller search trees. Extensive experiments on diverse, large-scale benchmarks demonstrate that KD-Club solves significantly more instances and usually with shorter running times than state-of-the-art solvers MADEC and KDBB, especially for larger and dense graphs. The approach offers robust performance across sparse and dense graphs and provides a framework that could be adapted to tighten upper bounds in other relaxation-based combinatorial problems.

Abstract

The Maximum k-Defective Clique Problem (MDCP) aims to find a maximum k-defective clique in a given graph, where a k-defective clique is a relaxation clique missing at most k edges. MDCP is NP-hard and finds many real-world applications in analyzing dense but not necessarily complete subgraphs. Exact algorithms for MDCP mainly follow the Branch-and-bound (BnB) framework, whose performance heavily depends on the quality of the upper bound on the cardinality of a maximum k-defective clique. The state-of-the-art BnB MDCP algorithms calculate the upper bound quickly but conservatively as they ignore many possible missing edges. In this paper, we propose a novel CoLoring-based Upper Bound (CLUB) that uses graph coloring techniques to detect independent sets so as to detect missing edges ignored by the previous methods. We then develop a new BnB algorithm for MDCP, called KD-Club, using CLUB in both the preprocessing stage for graph reduction and the BnB searching process for branch pruning. Extensive experiments show that KD-Club significantly outperforms state-of-the-art BnB MDCP algorithms on the number of solved instances within the cut-off time, having much smaller search tree and shorter solving time on various benchmarks.
Paper Structure (19 sections, 1 theorem, 1 equation, 2 figures, 2 tables, 5 algorithms)

This paper contains 19 sections, 1 theorem, 1 equation, 2 figures, 2 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Definitions
  4. Revisiting the Reduction Rules of KDBB
  5. Coloring-based Upper Bound
  6. Main Idea
  7. Definition of CLUB
  8. An Example for Illustration
  9. The ColorBound Algorithm
  10. Branch and Bound Algorithm
  11. General Framework
  12. Preprocessing Method
  13. Branch and Bound Process
  14. Empirical Evaluation
  15. Benchmark Datasets
  16. ...and 4 more sections

Key Result

Lemma 1

Suppose $C_i$ can be partitioned into $r_i$ independent sets $\{I_{i,1},\cdots,I_{i,r_i}\}$, adding any $1 \leq t \leq |C_i|$ vertices in $C_i$ to $S$ leads to at least $\mathcal{N}(t) = c\times \frac{d(d+1)}{2} + (r_i - c) \times \frac{d(d-1)}{2}$ more missing edges between the $t$ added vertices,

Figures (2)

  • Figure 1: An example for the upper bound calculation.
  • Figure 2: Ablation study on MDCP instances over all the 341 graphs with different $k$ values.

Theorems & Definitions (2)

  • Lemma 1
  • proof