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

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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 weakly robust polynomial time approximation scheme (PTAS) for UDGs expressed with edge-lengths that either (i) computes a clique partition or (ii) gives a certificate that the graph is not a UDG; for the case (i) it computes a clique partition having size that is guaranteed to be within (1+ε) of the optimum size 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 \(O(\frac{\log^{*} n}{{\varepsilon}^{O(1)}})\) 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 break down. Yet, surprisingly, it admits a (2+ε)-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 unweighted case for UDGs expressed in standard form.

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Authors and Affiliations

  1. Department of Computing Science, University of Alberta, Edmonton, Alberta, T6G 2E8, Canada

    Imran A. Pirwani & Mohammad R. Salavatipour

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  1. Imran A. Pirwani
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  2. Mohammad R. Salavatipour
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Correspondence to Imran A. Pirwani.

Additional information

A preliminary version of this article appeared in the Proceedings of the 12th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2010), Bergen, Norway, pp. 188–199 [21].

I.A. Pirwani was supported by Alberta Ingenuity.

M.R. Salavatipour was supported by NSERC and Alberta Ingenuity.

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Pirwani, I.A., Salavatipour, M.R. A Weakly Robust PTAS for Minimum Clique Partition in Unit Disk Graphs. Algorithmica 62, 1050–1072 (2012). https://doi.org/10.1007/s00453-011-9503-8

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