A Weakly-Robust PTAS for Minimum Clique Partition in Unit Disk Graphs

TL;DR

This paper presents a weakly-robust PTAS for minimum clique partition in unit disk graphs that either finds an approximate solution or certifies the graph is not a UDG, with improved approximation for the weighted case.

Contribution

It introduces a weakly-robust PTAS for UDGs, capable of handling edge-lengths and providing guarantees or certificates, and offers a new approximation for the weighted version.

Findings

PTAS guarantees a (1+ε)-approximation for UDGs.

Algorithm can be transformed into an efficient distributed PTAS.

Weighted case admits a (2+ε)-approximation, better than previous bounds.

Abstract

We consider the problem of partitioning the set of vertices of a given unit disk graph (UDG) into a minimum number of cliques. The problem is NP-hard and various constant factor approximations are known, with the current best ratio of 3. Our main result is a {\em weakly robust} polynomial time approximation scheme (PTAS) for UDGs expressed with edge-lengths, it either (i) computes a clique partition or (ii) gives a certificate that the graph is not a UDG; for the case (i) that it computes a clique partition, we show that it is guaranteed to be within $(1+\eps)$ ratio of the optimum if the input is UDG; however if the input is not a UDG it either computes a clique partition as in case (i) with no guarantee on the quality of the clique partition or detects that it is not a UDG. Noting that recognition of UDG's is NP-hard even if we are given edge lengths, our PTAS is a weakly-robust algorithm. Our algorithm can be transformed into an $O(\frac{\log^* n}{\eps^{O(1)}})$ time distributed PTAS. We consider a weighted version of the clique partition problem on vertex weighted UDGs that generalizes the problem. We note some key distinctions with the unweighted version, where ideas useful in obtaining a PTAS breakdown. Yet, surprisingly, it admits a $(2+\eps)$-approximation algorithm for the weighted case where the graph is expressed, say, as an adjacency matrix. This improves on the best known 8-approximation for the {\em unweighted} case for UDGs expressed in standard form.