Retraction: Improved Approximation Schemes for Dominating Set Problems in Unit Disk Graphs

TL;DR

This paper introduces faster polynomial-time approximation schemes for the Minimum Dominating Set and Minimum Connected Dominating Set problems in unit disk graphs, improving computational efficiency over previous methods.

Contribution

The paper presents two significantly faster PTAS algorithms for dominating set problems in unit disk graphs, utilizing improved dynamic programming techniques based on problem widths.

Findings

Achieved exponential speedup over previous PTAS algorithms.

Developed dynamic programming algorithms dependent on problem widths.

Provided approximation solutions within (1+ε) factor in polynomial time.

Abstract

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 $n^{O(1/\epsilon)}$ 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 $(1+\epsilon)$-approximate solutions to the Minimum Dominating Set (MDS) and the Minimum Connected Dominating Set (MCDS) of the input graph run in time $n^{O(1/\epsilon)}$. This can be compared to the best known $n^{O(1/\epsilon \log {1/\epsilon})}$-time PTAS by Nieberg and Hurink~[WAOA'05] for MDS that only uses graph structures and an $n^{O(1/\epsilon^2)}$-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.