A Weakly Robust 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 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 Graph Separator Line Optimal Partition Weighted Version Separation Theorem
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