Computational Study on a PTAS for Planar Dominating Set Problem

Marjan Marzban,Qian-Ping Gu

doi:10.3390/a6010043

Abstract

The dominating set problem is a core NP-hard problem in combinatorial optimization and graph theory, and has many important applications. Baker [JACM 41,1994] introduces a k-outer planar graph decomposition-based framework for designing polynomial time approximation scheme (PTAS) for a class of NP-hard problems in planar graphs. It is mentioned that the framework can be applied to obtain an O(2ckn) time, c is a constant, (1+1/k)-approximation algorithm for the planar dominating set problem. We show that the approximation ratio achieved by the mentioned application of the framework is not bounded by any constant for the planar dominating set problem. We modify the application of the framework to give a PTAS for the planar dominating set problem. With k-outer planar graph decompositions, the modified PTAS has an approximation ratio (1 + 2/k). Using 2k-outer planar graph decompositions, the modified PTAS achieves the approximation ratio (1+1/k) in O(22ckn) time. We report a computational study on the modified PTAS. Our results show that the modified PTAS is practical.

Highlights

An important research area in graph theory and networks is domination; it has been energetically investigated for many years due to its large number of real-world applications, such as resourceAlgorithms 2013, 6 allocation [1,2] and voting [3]
We show that the modified application of the framework gives a polynomial time approximation scheme (PTAS) with approximation ratio (1 + 2/k) for the planar dominating set problem
We study the practical performance of the PTAS for the planar dominating set problem

Summary

Introduction

An important research area in graph theory and networks is domination; it has been energetically investigated for many years due to its large number of real-world applications, such as resource. Given a graph G and a branch-decomposition of G with width β, FT algorithm finds an optimal solution in O(2(3 log4 3)β n) time for the dominating set problem. We show that the approximation ratio of Baker’s framework is not bounded by any constant for the planar dominating set problem when two “neighbor” k-outer planar subgraphs share only “one-level”. By decomposing G into 2k-outer planar subgraphs with “two-level” overlapping vertices, the modified PTAS achieves the approximation ratio (1 + 1/k) in O(22ck n) time.

Preliminaries

Baker’s Framework for Minimization Problem

Modified Framework for Planar Dominating Set Problem

Computational Study of PTAS

Concluding Remarks