Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An Effective Branch-and-Bound Algorithm with New Bounding Methods for the Maximum $s$-Bundle Problem

Jinghui Xue, Jiongzhi Zheng, Mingming Jin, Kun He

TL;DR

This work tackles the NP-hard Maximum $s$-Bundle Problem (MBP) by introducing a Partition-based Upper Bound (PUB) and a random-walk–based initial lower bound, forming the backbone of a new SCP branch-and-bound algorithm. PUB tightens upper bounds by partitioning the graph into $s$-components and aggregating bounds, while the random-walk LB yields higher-quality starting solutions. The SCP framework integrates PUB in both preprocessing and search, and demonstrates superior performance over state-of-the-art MBP solvers across diverse benchmarks, with strong ablation results and a demonstrated generalization to the Maximum $s$-Defective Clique Problem (MDCP). These advances enable more effective pruning and reduction, translating to drastically smaller search trees and faster solution times in practice. The methods offer practical impact for large, sparse networks and provide a pathway for extending LB techniques to related relaxation-clique problems.

Abstract

The Maximum s-Bundle Problem (MBP) addresses the task of identifying a maximum s-bundle in a given graph. A graph G=(V, E) is called an s-bundle if its vertex connectivity is at least |V|-s, where the vertex connectivity equals the minimum number of vertices whose deletion yields a disconnected or trivial graph. MBP is NP-hard and holds relevance in numerous realworld scenarios emphasizing the vertex connectivity. Exact algorithms for MBP mainly follow the branch-and-bound (BnB) framework, whose performance heavily depends on the quality of the upper bound on the cardinality of a maximum s-bundle and the initial lower bound with graph reduction. In this work, we introduce a novel Partition-based Upper Bound (PUB) that leverages the graph partitioning technique to achieve a tighter upper bound compared to existing ones. To increase the lower bound, we propose to do short random walks on a clique to generate larger initial solutions. Then, we propose a new BnB algorithm that uses the initial lower bound and PUB in preprocessing for graph reduction, and uses PUB in the BnB search process for branch pruning. Extensive experiments with diverse s values demonstrate the significant progress of our algorithm over state-of-the-art BnB MBP algorithms. Moreover, our initial lower bound can also be generalized to other relaxation clique problems.

An Effective Branch-and-Bound Algorithm with New Bounding Methods for the Maximum $s$-Bundle Problem

TL;DR

This work tackles the NP-hard Maximum -Bundle Problem (MBP) by introducing a Partition-based Upper Bound (PUB) and a random-walk–based initial lower bound, forming the backbone of a new SCP branch-and-bound algorithm. PUB tightens upper bounds by partitioning the graph into -components and aggregating bounds, while the random-walk LB yields higher-quality starting solutions. The SCP framework integrates PUB in both preprocessing and search, and demonstrates superior performance over state-of-the-art MBP solvers across diverse benchmarks, with strong ablation results and a demonstrated generalization to the Maximum -Defective Clique Problem (MDCP). These advances enable more effective pruning and reduction, translating to drastically smaller search trees and faster solution times in practice. The methods offer practical impact for large, sparse networks and provide a pathway for extending LB techniques to related relaxation-clique problems.

Abstract

The Maximum s-Bundle Problem (MBP) addresses the task of identifying a maximum s-bundle in a given graph. A graph G=(V, E) is called an s-bundle if its vertex connectivity is at least |V|-s, where the vertex connectivity equals the minimum number of vertices whose deletion yields a disconnected or trivial graph. MBP is NP-hard and holds relevance in numerous realworld scenarios emphasizing the vertex connectivity. Exact algorithms for MBP mainly follow the branch-and-bound (BnB) framework, whose performance heavily depends on the quality of the upper bound on the cardinality of a maximum s-bundle and the initial lower bound with graph reduction. In this work, we introduce a novel Partition-based Upper Bound (PUB) that leverages the graph partitioning technique to achieve a tighter upper bound compared to existing ones. To increase the lower bound, we propose to do short random walks on a clique to generate larger initial solutions. Then, we propose a new BnB algorithm that uses the initial lower bound and PUB in preprocessing for graph reduction, and uses PUB in the BnB search process for branch pruning. Extensive experiments with diverse s values demonstrate the significant progress of our algorithm over state-of-the-art BnB MBP algorithms. Moreover, our initial lower bound can also be generalized to other relaxation clique problems.
Paper Structure (20 sections, 3 theorems, 2 figures, 5 tables, 4 algorithms)

This paper contains 20 sections, 3 theorems, 2 figures, 5 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Definitions
  4. Prerequisites
  5. Identifying an s-Bundle
  6. Paritition-based Upper Bound
  7. Partition-based Bounding Rule
  8. Calculation of the PUB
  9. The Proposed SCP Algorithm
  10. Main Framework
  11. Lower Bound Initialization
  12. Preprocessing
  13. The Branch-and-Bound Process
  14. Experimental Results
  15. Benchmark Datasets
  16. ...and 5 more sections

Key Result

Lemma 1

If $S\subseteq V$ can induce an $s$-bundle in $G=(V,E)$, then for any $\mathcal{P} \subseteq S$, $\mathcal{P}$ can still induce an $s$-bundle.

Figures (2)

  • Figure 1: Partitions for the maximum $3$-bundle problem.
  • Figure 2: Ablation studies on the RealWorld and DIMACS2 benchmarks.

Theorems & Definitions (6)

  • Definition 1: $s$-component
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • proof
  • proof