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Title of Journal: Algorithmica

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Abbravation: Algorithmica

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Springer-Verlag

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10.1016/0959-8049(94)00360-h

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1432-0541

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A Weakly Robust PTAS for Minimum Clique Partition

Authors: Imran A Pirwani Mohammad R Salavatipour
Publish Date: 2011/03/17
Volume: 62, Issue: 3-4, Pages: 1050-1072
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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 NPhard 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 edgelengths 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 NPhard even if we are given edge lengths our PTAS is a weaklyrobust algorithm Our algorithm can be transformed into an Ofraclog nvarepsilonO1 time distributed PTASWe consider a weighted version of the clique partition problem on vertexweighted 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+εapproximation algorithm for the weighted case where the graph is expressed say as an adjacency matrix This improves on the best known 8approximation for the unweighted case for UDGs expressed in standard form

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