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Maximum Edge-based Quasi-Clique: Novel Iterative Frameworks

Hongbo Xia, Shengxin Liu, Zhaoquan Gu

TL;DR

The paper tackles the NP-hard MaxEQC problem, which seeks the largest edge-based γ-quasi-clique in a graph despite the lack of hereditary structure. It introduces an iterative paradigm that recasts MaxEQC as a sequence of hereditary subproblems, enabling stronger pruning and a novel bottom-up doubling strategy that reduces iterations to O(log s^*). Coupled with dynamic heuristics (EQC-Heu-Pro) and memoized solve-defect calls, the final algorithm EQC-Pro achieves substantial practical gains, outperforming state-of-the-art baselines by up to four orders of magnitude on 253 real-world graphs. The approach yields a scalable, exact solution framework with theoretical improvements in complexity and demonstrated empirical impact for large dense subgraph discovery, with potential extensions to weighted graphs in future work.

Abstract

Extracting cohesive subgraphs from complex networks is a fundamental task in graph analytics and is essential for understanding biological, social, and web graphs. The edge-based $γ$-quasi-clique model offers a flexible alternative by identifying subgraphs whose edge densities exceed a specified threshold $γ$. However, finding the exact maximum edge-based quasi-clique is computationally challenging, as the problem is NP-hard and lacks the hereditary property. These characteristics limit the effectiveness of conventional pruning methods and the development of efficient reduction rules. As a result, existing algorithms, such as QClique and FPCE, struggle to scale to large graphs. In this paper, we revisit the problem and propose a novel iterative framework that reformulates the problem as a sequence of hereditary subproblems, enabling more effective pruning and reduction strategies and improving the worst-case time complexity. Furthermore, we redesign the iterative process and introduce a novel heuristic to further improve practical efficiency. Extensive experiments on 253 large-scale real-world graphs demonstrate that our proposed algorithm EQC-Pro outperforms existing methods by up to four orders of magnitude.

Maximum Edge-based Quasi-Clique: Novel Iterative Frameworks

TL;DR

The paper tackles the NP-hard MaxEQC problem, which seeks the largest edge-based γ-quasi-clique in a graph despite the lack of hereditary structure. It introduces an iterative paradigm that recasts MaxEQC as a sequence of hereditary subproblems, enabling stronger pruning and a novel bottom-up doubling strategy that reduces iterations to O(log s^*). Coupled with dynamic heuristics (EQC-Heu-Pro) and memoized solve-defect calls, the final algorithm EQC-Pro achieves substantial practical gains, outperforming state-of-the-art baselines by up to four orders of magnitude on 253 real-world graphs. The approach yields a scalable, exact solution framework with theoretical improvements in complexity and demonstrated empirical impact for large dense subgraph discovery, with potential extensions to weighted graphs in future work.

Abstract

Extracting cohesive subgraphs from complex networks is a fundamental task in graph analytics and is essential for understanding biological, social, and web graphs. The edge-based -quasi-clique model offers a flexible alternative by identifying subgraphs whose edge densities exceed a specified threshold . However, finding the exact maximum edge-based quasi-clique is computationally challenging, as the problem is NP-hard and lacks the hereditary property. These characteristics limit the effectiveness of conventional pruning methods and the development of efficient reduction rules. As a result, existing algorithms, such as QClique and FPCE, struggle to scale to large graphs. In this paper, we revisit the problem and propose a novel iterative framework that reformulates the problem as a sequence of hereditary subproblems, enabling more effective pruning and reduction strategies and improving the worst-case time complexity. Furthermore, we redesign the iterative process and introduce a novel heuristic to further improve practical efficiency. Extensive experiments on 253 large-scale real-world graphs demonstrate that our proposed algorithm EQC-Pro outperforms existing methods by up to four orders of magnitude.
Paper Structure (19 sections, 6 theorems, 5 figures, 1 table, 9 algorithms)

This paper contains 19 sections, 6 theorems, 5 figures, 1 table, 9 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A Top-down Iterative Baseline Framework
  4. An Advanced Bottom-up Iterative Framework
  5. Our Novel Heuristic Strategies
  6. Basic Heuristic Methods
  7. Our Dynamic Refined Heuristic Strategies
  8. Our Final Iterative Algorithm: EQC-Pro
  9. Experiments
  10. Comparison with Existing Algorithms
  11. Comparison with Our Variants
  12. Related Work
  13. Conclusion
  14. Omitted Details of Top-down Iterative Framework in Section \ref{['sec:top-down']}
  15. Our Top-down Iterative Baseline EQC-TD
  16. ...and 4 more sections

Key Result

Proposition 1

Let $G=(V,E)$ be a graph and $S\subseteq V$ such that $G[S]$ is a $\gamma$-EQC. If there exists a vertex $v\in S$ with degree less than the average degree in $G[S]$, i.e., $d_G^S(v) \;\le\; \frac{2\,|E(G[S])|}{|S|}$, then $G[S']$, where $S' = S\setminus\{v\}$, is a $\gamma$‐EQC.

Figures (5)

  • Figure 1: Illustration of iterative frameworks.
  • Figure 2: Number of solved instances with varying $\gamma$.
  • Figure 3: Number of solved instances on Facebook.
  • Figure 4: Number of solved instances on Real-World.
  • Figure 5: Scalability test on instance socfb-konect.

Theorems & Definitions (8)

  • Definition 1
  • Proposition 1: Quasi‐hereditary property pattillo2013maximum
  • Definition 2: yu2006defective
  • Proposition 2
  • Lemma 1
  • Lemma 2
  • Proposition 3
  • Lemma 3