Minimum clique partition in unit disk graphs

TL;DR

This paper introduces two improved algorithms for the minimum clique partition problem in unit disk graphs, including a faster PTAS and a more accurate randomized algorithm, advancing the state of approximation methods.

Contribution

It presents a faster PTAS with reduced running time and a new randomized algorithm with a better approximation ratio for MCP in unit disk graphs.

Findings

A PTAS with runtime $n^{O(1/\\eps^2)}$

A randomized algorithm with approximation ratio 2.16

Improved approximation algorithms over previous methods

Abstract

The minimum clique partition (MCP) problem is that of partitioning the vertex set of a given graph into a minimum number of cliques. Given $n$ points in the plane, the corresponding unit disk graph (UDG) has these points as vertices, and edges connecting points at distance at most~1. MCP in unit disk graphs is known to be NP-hard and several constant factor approximations are known, including a recent PTAS. We present two improved approximation algorithms for minimum clique partition in unit disk graphs: (I) A polynomial time approximation scheme (PTAS) running in time $n^{O(1/\eps^2)}$. This improves on a previous PTAS with $n^{O(1/\eps^4)}$ running time \cite{PS09}. (II) A randomized quadratic-time algorithm with approximation ratio 2.16. This improves on a ratio 3 algorithm with $O(n^2)$ running time \cite{CFFP04}.