Graph Profiling for Vertex Cover: Targeted Reductions in a Branch and Reduce Solver

TL;DR

This paper investigates how graph characteristics influence the effectiveness of reduction rules in a branch and reduce solver for vertex cover, providing insights and tools to optimize solver performance based on measurable graph features.

Contribution

It introduces an enhanced solver, analyzes reduction performance relative to graph features, and proposes hypotheses and measures for selecting optimal reductions.

Findings

Certain reductions are more effective depending on graph density and degree distribution.

More reductions do not always improve runtime performance.

Structural graph characteristics can predict the most beneficial reductions.

Abstract

Akiba and Iwata [TCS, 2016] demonstrated that a branch and reduce (B&R) solver for the vertex cover problem can compete favorably with integer linear programming solvers (e.g., CPLEX). Our research question is are there graph characteristics that determine which reductions will be most effective? Not only is the answer affirmative, but relevant characteristics are easy to identify. To explore our ideas, we provide an enhanced version of the Akiba-Iwata solver that can (a) be configured with any subset of reductions and lower bounds; (b) print statistics such as time taken and number of vertices reduced by each reduction. Based on extensive experiments with benchmark and random instances we demonstrate that (i) more reductions do not necessarily lead to better runtimes; (ii) the subset of reductions leading to the best (or nearly the best) runtime can be predicted based on measurable characteristics of a graph, e.g., density and degree distribution; and (iii) exceptions have structural characteristics known in advance. Our primary contributions are 1. A thorough examination reduction routine performance in the context of graph characteristics. 2. Three primary hypotheses suggesting simple suites of reductions as the most efficient options. 3. Experiments with a large corpus of data to validate our hypotheses. 4. Measures that quantify a problem instance on two key dimensions to make our hypotheses concrete. 5. An enhanced open-source version of the Akiba-Iwata solver that enables our investigations and creates opportunities for future exploration. Our main objective is to provide guidance to a user so that, faced with a given problem instance or set of instances, they may most effectively use the available reductions. Ultimately these efforts can lead to an automated process.