The Generalized Independent and Dominating Set Problems on Unit Disk Graphs

TL;DR

This paper investigates generalized independent and dominating set problems on unit disk graphs, proving NP-hardness and providing polynomial-time approximation algorithms and PTAS for these problems.

Contribution

It introduces the maximum and minimum d-distance set problems on unit disk graphs, establishing their NP-hardness and offering approximation solutions.

Findings

NP-hardness of the generalized problems

Polynomial-time constant-factor approximation algorithms

PTAS for both problems

Abstract

In this article, we study a generalized version of the maximum independent set and minimum dominating set problems, namely, the maximum $d$-distance independent set problem and the minimum $d$-distance dominating set problem on unit disk graphs for a positive integer $d>0$. We first show that the maximum $d$-distance independent set problem and the minimum $d$-distance dominating set problem belongs to NP-hard class. Next, we propose a simple polynomial-time constant-factor approximation algorithms and PTAS for both the problems.