A performance study of some approximation algorithms for minimum dominating set in a graph

TL;DR

This study compares the performance of various approximation algorithms for the minimum dominating set problem, highlighting the efficiency and accuracy of LP rounding, greedy, and hybrid methods on different graph types.

Contribution

It introduces a hybrid algorithm combining greedy and LP rounding techniques and evaluates their performance on large and real-world graphs.

Findings

LP rounding outperforms others on sparse real-world graphs.

Hybrid algorithm excels on synthetic graphs and large problems.

Greedy algorithm performs best on hypercubes and k-Queens graphs.

Abstract

We implement and test the performances of several approximation algorithms for computing the minimum dominating set of a graph. These algorithms are the standard greedy algorithm, the recent LP rounding algorithms and a hybrid algorithm that we design by combining the greedy and LP rounding algorithms. All algorithms perform better than anticipated in their theoretical analysis, and have small performance ratios, measured as the size of output divided by the LP objective lower-bound. However, each may have advantages over the others. For instance, LP rounding algorithm normally outperforms the other algorithms on sparse real-world graphs. On a graph with 400,000+ vertices, LP rounding took less than 15 seconds of CPU time to generate a solution with performance ratio 1.011, while the greedy and hybrid algorithms generated solutions of performance ratio 1.12 in similar time. For synthetic graphs, the hybrid algorithm normally outperforms the others, whereas for hypercubes and k-Queens graphs, greedy outperforms the rest. Another advantage of the hybrid algorithm is to solve very large problems where LP solvers crash, as demonstrated on a real-world graph with 7.7 million+ vertices.