Constant-Time Approximation Algorithms for the Optimum Branching Problem on Sparse Graphs

2012 ◽  
Author(s):  
Mitsuru Kusumoto ◽  
Yuichi Yoshida ◽  
Hiro Ito
Keyword(s):  
Approximation Algorithms ◽  
Constant Time ◽  
Sparse Graphs ◽  
Time Approximation ◽  
Branching Problem

scholarly journals A Performance Study of Some Approximation Algorithms for Computing a Small Dominating Set in a Graph

Algorithms ◽  
2020 ◽  
Vol 13 (12) ◽  
pp. 339
Author(s):  
Jonathan Li ◽  
Rohan Potru ◽  
Farhad Shahrokhi
Keyword(s):  
Approximation Algorithms ◽  
Real World ◽  
Hybrid Algorithm ◽  
Dominating Set ◽  
Performance Ratio ◽  
Performance Study ◽  
Similar Time ◽  
Sparse Graphs ◽  
Lp Rounding ◽  
Better Than

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.

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