Two New Upper Bounds for the Maximum k-plex Problem

TL;DR

This paper introduces two novel upper bounds, RelaxGCB and RelaxPUB, for the Maximum k-plex Problem, enhancing branch-and-bound algorithms and demonstrating superior performance on diverse graph instances.

Contribution

The paper proposes two new upper bounds, RelaxGCB and RelaxPUB, improving the efficiency of exact algorithms for MKP by leveraging graph coloring and partition complementarity.

Findings

RelaxGCB significantly outperforms previous coloring-based bounds.

RelaxPUB combines RelaxGCB and partition bounds for enhanced upper bounds.

Extensive experiments show improved performance and robustness of the new algorithms.

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.