A PTAS for Minimum Clique Partition in Unit Disk Graphs

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 polynomial time approximation scheme(PTAS) for this problem on UDG. In fact, we present a robust algorithm that given a graph $G$ (not necessarily UDG) 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 robust algorithm. Our main technical contribution involves showing the property of {\em separability} of an optimal clique partition; that there exists an optimal clique partition where the convex hulls of the cliques are pairwise non-overlapping. Our algorithm can be transformed into an $O(\frac{\log^*n}{\eps^{O(1)}})$ time distributed polynomial-time approximation scheme (PTAS). We also consider a weighted version of the clique partition problem on vertex weighted UDGs; we show that ideas developed for the unweighted case do not help. Yet we show that the problem admits a $(2+\eps)$-approximation algorithm.