[Paper Review] An Exact Algorithm for Minimum Weight Vertex Cover Problem in Large Graphs

Luzhi Wang, Chu-Min Li+3 more|arXiv (Cornell University)|Mar 14, 2019

Advanced Graph Theory Research24 references9 citations

TL;DR

This paper presents BMWVC, an exact branch-and-bound algorithm for the minimum weight vertex cover problem (MWVC) in large, sparse graphs. It introduces four novel graph reduction rules, a tight lower bound based on disjoint cliques, and an efficient branching heuristic, enabling BMWVC to outperform both exact and heuristic solvers on real-world large graphs by achieving optimal solutions faster and more consistently.

ABSTRACT

This paper proposes a novel branch-and-bound(BMWVC) algorithm to exactly solve the minimum weight vertex cover problem (MWVC) in large graphs. The original contribution is several new graph reduction rules, allowing to reduce a graph G and the time needed to find a minimum weight vertex cover in G. Experiments on large graphs from real-world applications show that the reduction rules are effective and the resulting BMWVC algorithm outperforms relevant exact and heuristic MWVC algorithms.

Motivation & Objective

• To address the lack of practical exact solvers for the minimum weight vertex cover problem (MWVC) in large real-world graphs.
• To design an algorithm specifically tailored for MWVC, overcoming limitations of existing FPT algorithms and SAT/MWC-based approaches.
• To develop effective reduction rules that simplify large sparse graphs without losing optimality guarantees.
• To improve search efficiency through a tight lower bound and an effective branching heuristic.
• To evaluate BMWVC on real-world and application-specific large graphs, demonstrating its superiority over state-of-the-art exact and heuristic solvers.

Proposed method

• Proposes four novel graph reduction rules that simplify the MWVC instance by removing or fixing vertices based on degree, weight, and neighborhood structure.
• Employs a lower bound computed from disjoint cliques in the graph to prune non-promising branches during search.
• Uses a heuristic for vertex selection in the branch-and-bound process that prioritizes vertices likely to reduce the search space efficiently.
• Applies the branch-and-bound framework to systematically explore the solution space while leveraging reductions and pruning to improve performance.
• Operates directly on the original graph without transforming to complementary graphs or encoding into SAT, avoiding scalability issues from dense complementary graphs.
• Implements a memory-aware strategy that stops on memory overflow and reports the best solution found, with optimality proven only when memory allows.

Experimental results

Research questions

• RQ1 Can novel graph reduction rules significantly reduce the size of large, sparse vertex-weighted graphs while preserving the optimality of the MWVC solution?
• RQ2 Can a tight lower bound based on disjoint cliques effectively prune large portions of the search space in MWVC?
• RQ3 Does an efficient branching heuristic outperform random or degree-based selection in reducing the runtime of exact MWVC solvers?
• RQ4 Can an exact MWVC solver designed specifically for MWVC outperform general-purpose solvers like SAT encodings or MWC solvers on real-world large graphs?
• RQ5 How does BMWVC compare in performance and solution quality against state-of-the-art heuristic solvers (e.g., Dynwvc1/2) and exact solvers (e.g., WLMC, SBMS) on large and application-specific graphs?

Key findings

• BMWVC proved optimal solutions for 46 out of 47 real-world large graphs with edge density < 7, with only one instance failing due to memory overflow.
• On 17 graphs, BMWVC found and proved the optimal solution faster than the best heuristic solvers Dynwvc1 and Dynwvc2, which failed to find the optimum on 16 instances.
• BMWVC solved 255 out of 293 application graphs (from WDP, KES, ECC, REF) within 5 hours, outperforming SBMS (86 solved), Dynwvc1/Dynwvc2 (209 solved), and coming close to WLMC (269 solved).
• The proposed reduction rules were highly effective, reducing massive graphs to tractable sizes, enabling exact solving where previous methods failed.
• The tight lower bound based on disjoint cliques significantly pruned the search space, contributing to faster convergence.
• BMWVC is the first exact solver specifically designed for MWVC in large graphs, avoiding the inefficiency of SAT or MWC solver transformations.