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تسجيل مستخدم جديدThe Generalized Independent and Dominating Set Problems on Unit Disk Graphs
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Sangram Kishor Jena Mr
تاريخ النشر
2020
مجال البحث
الهندسة المعلوماتية
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English
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In this article, we study a generalized version of the maximum independent set and minimum dominating set problems, namely, the maximum $d$-distance independent set problem and the minimum $d$-distance dominating set problem on unit disk graphs for a positive integer $d>0$. We first show that the maximum $d$-distance independent set problem and the minimum $d$-distance dominating set problem belongs to NP-hard class. Next, we propose a simple polynomial-time constant-factor approximation algorithms and PTAS for both the problems.
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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 $n^{O(1/epsilon)}$ 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 PTASs for dominating set problems in unit disk graphs. Given a geometric representation of a unit disk graph, our PTASs 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~[WAOA05] 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 Optim09]. Our key ingredients are improved dynamic programming algorithms that depend exponentially on more essential 1-dimensional widths of the problems.
We recently introduced the graph invariant twin-width, and showed that first-order model checking can be solved in time $f(d,k)n$ for $n$-vertex graphs given with a witness that the twin-width is at most $d$, called $d$-contraction sequence or $d$-se
quence, and formulas of size $k$ [Bonnet et al., FOCS 20]. The inevitable price to pay for such a general result is that $f$ is a tower of exponentials of height roughly $k$. In this paper, we show that algorithms based on twin-width need not be impractical. We present $2^{O(k)}n$-time algorithms for $k$-Independent Set, $r$-Scattered Set, $k$-Clique, and $k$-Dominating Set when an $O(1)$-sequence is provided. We further show how to solve weighted $k$-Independent Set, Subgraph Isomorphism, and Induced Subgraph Isomorphism, in time $2^{O(k log k)}n$. These algorithms are based on a dynamic programming scheme following the sequence of contractions forward. We then show a second algorithmic use of the contraction sequence, by starting at its end and rewinding it. As an example, we establish that bounded twin-width classes are $chi$-bounded. This significantly extends the $chi$-boundedness of bounded rank-width classes, and does so with a very concise proof. The third algorithmic use of twin-width builds on the second one. Playing the contraction sequence backward, we show that bounded twin-width graphs can be edge-partitioned into a linear number of bicliques, such that both sides of the bicliques are on consecutive vertices, in a fixed vertex ordering. Given that biclique edge-partition, we show how to solve the unweighted Single-Source Shortest Paths and hence All-Pairs Shortest Paths in sublinear time $O(n log n)$ and time $O(n^2 log n)$, respectively. Finally we show that Min Dominating Set and related problems have constant integrality gaps on bounded twin-width classes, thereby getting constant approximations on these classes.
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