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Maximal Clique Enumeration with Hybrid Branching and Early Termination

Kaixin Wang, Kaiqiang Yu, Cheng Long

TL;DR

This work addresses maximal clique enumeration (MCE) by introducing HBBMC, a hybrid branch-and-bound framework that merges vertex-oriented BK branching with edge-oriented BK branching to enable stronger pruning. It further enhances efficiency with an early termination technique that directly constructs maximal cliques from dense subgraphs using topological information, applicable across BB frameworks. Theoretical results show a worst-case time bound of $O(\delta m + \tau m \cdot 3^{\tau/3})$, tightening performance under realistic conditions where $\delta \ge \tau + \frac{3}{\ln 3}\ln \rho$, and practical gains are demonstrated via extensive experiments on real and synthetic graphs. The combination of hybrid branching and topology-aware termination yields substantial speedups over state-of-the-art VBBMC-based methods and offers a scalable approach for MCE in large, real-world networks.

Abstract

Maximal clique enumeration (MCE) is crucial for tasks like community detection and biological network analysis. Existing algorithms typically adopt the branch-and-bound framework with the vertex-oriented Bron-Kerbosch (BK) branching strategy, which forms the sub-branches by expanding the partial clique with a vertex. In this paper, we present a novel approach called HBBMC, a hybrid framework combining vertex-oriented BK branching and edge-oriented BK branching, where the latter adopts a branch-and-bound framework which forms the sub-branches by expanding the partial clique with an edge. This hybrid strategy enables more effective pruning and helps achieve a worst-case time complexity better than the best known one under a condition that holds for the majority of real-world graphs. To further enhance efficiency, we introduce an early termination technique, which leverages the topological information of the graphs and constructs the maximal cliques directly without branching. Our early termination technique is applicable to all branch-and-bound frameworks. Extensive experiments demonstrate the superior performance of our techniques.

Maximal Clique Enumeration with Hybrid Branching and Early Termination

TL;DR

This work addresses maximal clique enumeration (MCE) by introducing HBBMC, a hybrid branch-and-bound framework that merges vertex-oriented BK branching with edge-oriented BK branching to enable stronger pruning. It further enhances efficiency with an early termination technique that directly constructs maximal cliques from dense subgraphs using topological information, applicable across BB frameworks. Theoretical results show a worst-case time bound of , tightening performance under realistic conditions where , and practical gains are demonstrated via extensive experiments on real and synthetic graphs. The combination of hybrid branching and topology-aware termination yields substantial speedups over state-of-the-art VBBMC-based methods and offers a scalable approach for MCE in large, real-world networks.

Abstract

Maximal clique enumeration (MCE) is crucial for tasks like community detection and biological network analysis. Existing algorithms typically adopt the branch-and-bound framework with the vertex-oriented Bron-Kerbosch (BK) branching strategy, which forms the sub-branches by expanding the partial clique with a vertex. In this paper, we present a novel approach called HBBMC, a hybrid framework combining vertex-oriented BK branching and edge-oriented BK branching, where the latter adopts a branch-and-bound framework which forms the sub-branches by expanding the partial clique with an edge. This hybrid strategy enables more effective pruning and helps achieve a worst-case time complexity better than the best known one under a condition that holds for the majority of real-world graphs. To further enhance efficiency, we introduce an early termination technique, which leverages the topological information of the graphs and constructs the maximal cliques directly without branching. Our early termination technique is applicable to all branch-and-bound frameworks. Extensive experiments demonstrate the superior performance of our techniques.

Paper Structure

This paper contains 20 sections, 8 theorems, 11 equations, 5 figures, 7 tables, 10 algorithms.

Table of Contents

  1. Introduction
  2. Problem and Preliminaries
  3. Branch-and-bound Algorithms
  4. The Existing Branch-and-bound Framework: VBBMC
  5. A New Branch-and-bound Framework: EBBMC
  6. A Hybrid Branch-and-bound Framework: HBBMC
  7. Early Termination Technique
  8. Enumerating Maximal Cliques from 2-Plex in Nearly Optimal Time
  9. Enumerating Maximal Cliques from 3-Plex in Nearly Optimal Time
  10. Experiments
  11. Experimental Setup
  12. Experimental Results on Real Datasets
  13. Related Work
  14. Conclusion
  15. Time Complexities of VBBMC Algorithms
  16. ...and 5 more sections

Key Result

Theorem 1

Given a graph $G=(V,E)$, the time and space complexities of EBBMC with the truss-based edge ordering are $O(\delta m +\tau m \cdot 2^{\tau})$ and $O(m+n)$, respectively.

Figures (5)

  • Figure 1: (a) and (b) show two possible cases to expand a partial maximal clique $S_p$ of a given path $p$ when $v_i\in S_p$ is the last vertex. (c) and (d) show three possible cases to enumerate all maximal cliques given a cycle $c$. Note that "Include" (resp. "Exclude") means that the vertices that are included in (resp. excluded from) a maximal clique.
  • Figure 2: A worst case for BK_Rcd algorithm.
  • Figure 3: Example of a 2-plex and its inverse graph.
  • Figure 4: Example of a 3-plex and its inverse graph.
  • Figure 5: Results on synthetic datasets.

Theorems & Definitions (17)

  • Definition 1: Clique erdos1935combinatorial
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • ...and 7 more