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 Linear programming (LP) rounding algorithms and a hybrid algorithm that we design by combining the greedy and LP rounding algorithms. Over the range of test data, all algorithms perform better than anticipated in theory, and have small performance ratios, measured as the size of output divided by the LP objective lower bound. However, each 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 s 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 that are suitable for application of LP rounding (sparse graphs) but LP formulations become formidable in practice and LP solvers crash, as we observed on a real-world graph with 7.7 million+ vertices and a planar graph on 1,000,000 vertices.
Highlights
Domination theory has its roots in the k-Queens problem in the 18th century
All algorithms perform better than anticipated in theory, with respect to the performance ratios, measured as the value of solution divided by the computed Linear programming (LP) objective lower bound
On a graph with 400,000+ vertices, LP rounding took less than 15 s 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
Summary
Domination theory has its roots in the k-Queens problem in the 18th century. Later, in 1957, Berge [1] formally introduced the domination number of a graph. No algorithm for approximating γ( G ) can improve the asymptotic worst-case performance ratio of the greedy algorithm Different variations of this algorithm are proposed and some are tested in practice. For graphs with bounded arboricity, one can improve the logarithmic performance ratio of the greedy algorithm to a constant. The recent LP rounding methods referenced above have a bounded performance ratio, which is better than greedy, but to our knowledge, and in contrast to the greedy algorithm, the performance of the LP-based approaches have not been tested in practice. Our hybrid algorithm first solves the problem using the greedy algorithm and finds a dominating set, takes a portion of vertices in this set and forces their values to be 1 in the linear programming formulation, solves the resulting linear program, and properly rounds the solution
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
