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Two New Upper Bounds for the Maximum k-plex Problem

Jiongzhi Zheng, Mingming Jin, Kun He

TL;DR

This work tackles the NP-hard Maximum k-plex Problem (MKP) within a branch-and-bound framework by strengthening two major upper-bounds: the graph coloring-based bound (GCB) and the partition-based bound (PUB). The authors introduce RelaxGCB, which tightens GCB by incorporating connectivity to the current partial k-plex and relaxing the independent-set structure, and RelaxPUB, a novel融合 that iteratively combines RelaxGCB with DisePUB to exploit complementarity between the two bound families. They implement eight new MKP solvers by replacing the original bounds with RelaxGCB and RelaxPUB, and demonstrate, across dense and large-sparse graphs and multiple k-values, substantial pruning improvements and higher solve rates. The results indicate that RelaxGCB and especially RelaxPUB provide robust, generalizable performance gains, enabling faster and more scalable MKP solving in practical network analysis tasks.

Abstract

A k-plex in a graph is a vertex set where each vertex is non-adjacent to at most k vertices (including itself) in this set, and the Maximum k-plex Problem (MKP) is to find the largest k-plex in the graph. As a practical NP-hard problem, MKP has many important real-world applications, such as the analysis of various complex networks. Branch-and-bound (BnB) algorithms are a type of well-studied and effective exact algorithms for MKP. Recent BnB MKP algorithms involve two kinds of upper bounds based on graph coloring and partition, respectively, that work in different perspectives and thus are complementary with each other. In this paper, we first propose a new coloring-based upper bound, termed Relaxed Graph Color Bound (RelaxGCB), that significantly improves the previous coloring-based upper bound. We further propose another new upper bound, termed RelaxPUB, that incorporates RelaxGCB and a partition-based upper bound in a novel way, making use of their complementarity. We apply RelaxGCB and RelaxPUB to state-of-the-art BnB MKP algorithms and produce eight new algorithms. Extensive experiments using diverse k values on hundreds of instances based on dense and massive sparse graphs demonstrate the excellent performance and robustness of our proposed methods.

Two New Upper Bounds for the Maximum k-plex Problem

TL;DR

This work tackles the NP-hard Maximum k-plex Problem (MKP) within a branch-and-bound framework by strengthening two major upper-bounds: the graph coloring-based bound (GCB) and the partition-based bound (PUB). The authors introduce RelaxGCB, which tightens GCB by incorporating connectivity to the current partial k-plex and relaxing the independent-set structure, and RelaxPUB, a novel融合 that iteratively combines RelaxGCB with DisePUB to exploit complementarity between the two bound families. They implement eight new MKP solvers by replacing the original bounds with RelaxGCB and RelaxPUB, and demonstrate, across dense and large-sparse graphs and multiple k-values, substantial pruning improvements and higher solve rates. The results indicate that RelaxGCB and especially RelaxPUB provide robust, generalizable performance gains, enabling faster and more scalable MKP solving in practical network analysis tasks.

Abstract

A k-plex in a graph is a vertex set where each vertex is non-adjacent to at most k vertices (including itself) in this set, and the Maximum k-plex Problem (MKP) is to find the largest k-plex in the graph. As a practical NP-hard problem, MKP has many important real-world applications, such as the analysis of various complex networks. Branch-and-bound (BnB) algorithms are a type of well-studied and effective exact algorithms for MKP. Recent BnB MKP algorithms involve two kinds of upper bounds based on graph coloring and partition, respectively, that work in different perspectives and thus are complementary with each other. In this paper, we first propose a new coloring-based upper bound, termed Relaxed Graph Color Bound (RelaxGCB), that significantly improves the previous coloring-based upper bound. We further propose another new upper bound, termed RelaxPUB, that incorporates RelaxGCB and a partition-based upper bound in a novel way, making use of their complementarity. We apply RelaxGCB and RelaxPUB to state-of-the-art BnB MKP algorithms and produce eight new algorithms. Extensive experiments using diverse k values on hundreds of instances based on dense and massive sparse graphs demonstrate the excellent performance and robustness of our proposed methods.
Paper Structure (18 sections, 3 theorems, 5 figures, 1 table, 4 algorithms)

This paper contains 18 sections, 3 theorems, 5 figures, 1 table, 4 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Definitions
  4. Framework of BnB MKP Algorithms
  5. The RelaxGCB Bound
  6. A Tighter Upper Bound for Independent Sets
  7. Relax the Independent Sets
  8. An Example for Illustration
  9. The RelaxColoring Algorithm
  10. The RelaxPUB Bound
  11. Revisiting DisePUB
  12. Complementarity of GCB and PUB
  13. Combining RelaxGCB and DisePUB
  14. Experimental Results
  15. Experimental Setup
  16. ...and 3 more sections

Key Result

Lemma 1

Suppose $I = \{v_1, v_2, \cdots, v_{|I|}\} \subseteq C$ is an independent set and $\delta_k^{-}(S,v_1) \geq \delta_k^{-}(S,v_2) \geq \cdots \geq \delta_k^{-}(S,v_{|I|})$, $\mathop{\max}\{i | \delta_k^{-}(S,v_i) \geq i\}$ is an upper bound of the number of vertices that $I$ can provide for $S$.

Figures (5)

  • Figure 1: An example for comparing the upper bounds.
  • Figure 2: Two examples for demonstrating the complementarity.
  • Figure 3: Comparisons on the dense 2nd DIMACS benchmark. For the baselines, Maplex is based on GCB while the other three on PUB.
  • Figure 4: Comparisons on the sparse Real-world benchmark. For the baselines, Maplex is based on GCB while the other three on PUB
  • Figure 5: Ablation studies on each baseline over all the tested instances.

Theorems & Definitions (8)

  • Lemma 1: TISUB
  • proof
  • Definition 1: Conflict Vertex
  • Definition 2: Loose Vertex
  • Lemma 2
  • proof
  • Lemma 3
  • proof