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Efficient $k$-Clique Listing: An Edge-Oriented Branching Strategy

Kaixin Wang, Kaiqiang Yu, Cheng Long

TL;DR

The paper introduces EBBkC, a novel edge-oriented branch-and-bound framework for listing all $k$-cliques, which grows partial cliques by adding an edge rather than a single vertex. By employing edge orderings based on truss decomposition, colorings, and a hybrid approach, EBBkC achieves a theoretical bound of $O(\delta m + k m (\tau/2)^{k-2})$ with $\tau < \delta$, outperforming prior vertex-oriented methods for $k>3$. It further enhances efficiency with an early termination strategy that exploits dense subgraph structures (cliques, 2-plexes, and $t$-plexes) via combinatorial listing and inverse graphs. Extensive experiments on 19 real graphs demonstrate that EBBkC, especially with early termination, significantly exceeds the performance of state-of-the-art VBBkC-based algorithms. The approach also shows promise for extending to maximal/maximum/diversified clique problems and other dense subgraph mining tasks.

Abstract

$k$-clique listing is a vital graph mining operator with diverse applications in various networks. The state-of-the-art algorithms all adopt a branch-and-bound (BB) framework with a vertex-oriented branching strategy (called VBBkC), which forms a sub-branch by expanding a partial $k$-clique with a vertex. These algorithms have the time complexity of $O(k m (δ/2)^{k-2})$, where $m$ is the number of edges in the graph and $δ$ is the degeneracy of the graph. In this paper, we propose a BB framework with a new edge-oriented branching (called EBBkC), which forms a sub-branch by expanding a partial $k$-clique with two vertices that connect each other (which correspond to an edge). We explore various edge orderings for EBBkC such that it achieves a time complexity of $O(δm + k m (τ/2)^{k-2})$, where $τ$ is an integer related to the maximum truss number of the graph and we have $τ< δ$. The time complexity of EBBkC is better than that of VBBkC algorithms for $k>3$ since both $O(δm)$ and $O(k m (τ/2)^{k-2})$ are bounded by $O(k m (δ/2)^{k-2})$. Furthermore, we develop specialized algorithms for sub-branches on dense graphs so that we can early-terminate them and apply the specialized algorithms. We conduct extensive experiments on 19 real graphs, and the results show that our newly developed EBBkC-based algorithms with the early termination technique consistently and largely outperform the state-of-the-art (VBBkC-based) algorithms.

Efficient $k$-Clique Listing: An Edge-Oriented Branching Strategy

TL;DR

The paper introduces EBBkC, a novel edge-oriented branch-and-bound framework for listing all -cliques, which grows partial cliques by adding an edge rather than a single vertex. By employing edge orderings based on truss decomposition, colorings, and a hybrid approach, EBBkC achieves a theoretical bound of with , outperforming prior vertex-oriented methods for . It further enhances efficiency with an early termination strategy that exploits dense subgraph structures (cliques, 2-plexes, and -plexes) via combinatorial listing and inverse graphs. Extensive experiments on 19 real graphs demonstrate that EBBkC, especially with early termination, significantly exceeds the performance of state-of-the-art VBBkC-based algorithms. The approach also shows promise for extending to maximal/maximum/diversified clique problems and other dense subgraph mining tasks.

Abstract

-clique listing is a vital graph mining operator with diverse applications in various networks. The state-of-the-art algorithms all adopt a branch-and-bound (BB) framework with a vertex-oriented branching strategy (called VBBkC), which forms a sub-branch by expanding a partial -clique with a vertex. These algorithms have the time complexity of , where is the number of edges in the graph and is the degeneracy of the graph. In this paper, we propose a BB framework with a new edge-oriented branching (called EBBkC), which forms a sub-branch by expanding a partial -clique with two vertices that connect each other (which correspond to an edge). We explore various edge orderings for EBBkC such that it achieves a time complexity of , where is an integer related to the maximum truss number of the graph and we have . The time complexity of EBBkC is better than that of VBBkC algorithms for since both and are bounded by . Furthermore, we develop specialized algorithms for sub-branches on dense graphs so that we can early-terminate them and apply the specialized algorithms. We conduct extensive experiments on 19 real graphs, and the results show that our newly developed EBBkC-based algorithms with the early termination technique consistently and largely outperform the state-of-the-art (VBBkC-based) algorithms.
Paper Structure (21 sections, 7 theorems, 10 equations, 15 figures, 2 tables, 7 algorithms)

This paper contains 21 sections, 7 theorems, 10 equations, 15 figures, 2 tables, 7 algorithms.

Table of Contents

  1. Introduction
  2. Problem and Preliminaries
  3. The Branch-and-Bound Framework of Existing Algorithms: VBBkC
  4. A New Branch-and-Bound Framework: EBBkC
  5. Motivation and Overview of EBBkC
  6. EBBkC-T: EBBkC with the Truss-based Edge Ordering
  7. EBBkC-C: EBBkC with the Color-based Edge Ordering
  8. EBBkC-H: EBBkC with Hybrid Edge Ordering
  9. Other Potential Applications of EBBkC
  10. Early Termination Technique
  11. Listing $k$-Cliques from 2-Plex in Nearly Optimal Time
  12. Listing $k$-Cliques from $t$-Plex with $t\ge 3$
  13. Experiments
  14. Experimental Setup
  15. Experimental Results
  16. ...and 6 more sections

Key Result

lemma 1

When applying the truss-based edge ordering $\pi_\tau(g)$ at $(S,g,l)$, we have $\tau(g)<\delta(g)$, where $\delta(g)$ is the degeneracy of $g$.

Figures (15)

  • Figure 1: Illustration of VBBkC and EBBkC.
  • Figure 2: Color-based edge ordering and pruning rules.
  • Figure 3: Examples of a 2-plex and its inverse graph.
  • Figure 4: Comparison with baselines on the small-$\omega$ graphs, varying $k$ from 4 to $\omega$.
  • Figure 5: Comparison with baselines on the large-$\omega$ graphs, varying $k$ from 4 to 9 and from $\omega-5$ to $\omega$.
  • ...and 10 more figures

Theorems & Definitions (8)

  • Definition 2.1: $k$-clique erdos1935combinatorial
  • lemma 1
  • theorem 1
  • theorem 2
  • theorem 3
  • theorem 4
  • lemma 2
  • theorem 5