Retraction: Improved Approximation Schemes for Dominating Set Problems in Unit Disk Graphs
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Retraction note: After posting the manuscript on arXiv, we were informed by Erik Jan van Leeuwen that both results were known and they appeared in his thesis[vL09]. A PTAS for MDS is at Theorem 6.3.21 on page 79 and A PTAS for MCDS is at Theorem 6.3.31 on page 82. The techniques used are very similar. He noted that the idea for dealing with the connected version using a constant number of extra layers in the shifting technique not only appeared Zhang et al.[ZGWD09] but also in his 2005 paper [vL05]. Finally, van Leeuwen also informed us that the open problem that we posted has been resolved by Marx [Mar06, Mar07] who showed that an efficient PTAS for MDS does not exist [Mar06] and under ETH, the running time of n^O(1/ϵ) is best possible [Mar07]. We thank Erik Jan van Leeuwen for the information and we regret that we made this mistake. Abstract before retraction: We present two (exponentially) faster PTAS's for dominating set problems in unit disk graphs. Given a geometric representation of a unit disk graph, our PTAS's that find (1+ϵ)-approximate solutions to the Minimum Dominating Set (MDS) and the Minimum Connected Dominating Set (MCDS) of the input graph run in time n^O(1/ϵ). This can be compared to the best known n^O(1/ϵlog1/ϵ)-time PTAS by Nieberg and Hurink [WAOA'05] for MDS that only uses graph structures and an n^O(1/ϵ^2)-time PTAS for MCDS by Zhang, Gao, Wu, and Du [J Glob Optim'09]. Our key ingredients are improved dynamic programming algorithms that depend exponentially on more essential 1-dimensional "widths" of the problems.
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