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

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 best known ratio of 3. Our main result is a weakly robust polynomial time approximation scheme (PTAS) for UDGs expressed with edge-lengths and ε> 0 that either (i) computes a clique partition, or (ii) produces a certificate proving that the graph is not a UDG; if the graph is a UDG, then our partition is guaranteed to be within (1 + ε) ratio of the optimum; however, if the graph is not a UDG, it either computes a clique partition, or detects that the graph is not a UDG. Noting that recognition of UDG’s is NP-hard even with edge lengths, this is a significant weakening of the input model.We consider a weighted version of the problem on vertex weighted UDGs that generalizes the problem. We note some key distinctions with the unweighted version, where ideas crucial in obtaining a PTAS breakdown. Nevertheless, the weighted version admits a (2 + ε)-approximation algorithm even when the graph is expressed, say, as an adjacency matrix. This is an improvement on the best known 8-approximation for the unweighted case for UDGs expressed in standard form.KeywordsInput GraphSeparator LineOptimal PartitionWeighted VersionSeparation TheoremThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.