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A PTAS for Minimum Clique Partition in Unit Disk Graphs Open Access


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Pirwani, Imran
Salavatipour, Mohammad
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Distributed computation
Polynomial time approximation scheme
Unit disk graphs
Clique partition
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Computing Science Technical Report
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Technical report TR09-06. 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 + ε) 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 algorithm can be transformed into an O (log*n/εο(1))time distributed polynomial-time approximation scheme (PTAS). Finally, we consider a weighted version of the clique partition problem on vertex weighted UDGs; the weight of a clique is the weight of a heaviest vertex in it, and the weight of a clique partition is the sum of the weights of the cliques in it. This formulation generalizes the classical clique partition problem. We show that the problem admits a (2 + ε)-approximation algorithm for the weighted version of the problem where the graph is expressed in standard form, for example, as an adjacency matrix. This improves on the best known algorithm which constructs an 8-approximation for the unweighted case for UDGs expressed in standard form.
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