Improved Parameterized Algorithms for Planar Dominating Set

Kanj, Iyad A.; Perković, Ljubomir

doi:10.1007/3-540-45687-2_33

Cited by 42 publications

(36 citation statements)

References 8 publications

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“…The basic idea behind the algorithm is to apply a simple divide-and-conquer strategy by removing the vertices in the family F μ to split the graph into chunks, then to compute a minimum dominating set for the resulting chunks using the algorithm introduced in [33], which is a variation of Baker's algorithm [8]. To do this, for each vertex v in the F μ , we "guess" whether v is in the minimum dominating set for G or not (basically, what we mean by guessing is enumerating all sequences corresponding to the different possibilities).…”

Section: Is a Layer Decomposition Of G Then Clearly The Vertices In mentioning

confidence: 99%

“…For each guess of all the vertices in F μ , we will solve the corresponding instance with respect to that guess. It was shown in [33] how this guessing process can be achieved using at most three statuses per vertex. Hence, guessing the vertices in F μ can be done by enumerating at most 3 |Fμ| ≤ 3 67k/l ternary sequences.…”

Section: Is a Layer Decomposition Of G Then Clearly The Vertices In mentioning

confidence: 99%

“…After guessing each vertex in the separator and updating the graph accordingly, the instance becomes an instance of a variation of the minimum dominating set problem due to the constraints placed on some of the vertices in the graph. Kanj and Perković introduced an algorithm in [33], which is a variation of Baker's algorithm [8], to solve this problem. The algorithm introduced in [33] solves this problem on the chunks in time O(27 d+1 n), where d is the maximum number of layers in a chunk (i.e., the maximum depth of a chunk).…”

Section: Is a Layer Decomposition Of G Then Clearly The Vertices In mentioning

confidence: 99%

“…Kanj and Perković introduced an algorithm in [33], which is a variation of Baker's algorithm [8], to solve this problem. The algorithm introduced in [33] solves this problem on the chunks in time O(27 d+1 n), where d is the maximum number of layers in a chunk (i.e., the maximum depth of a chunk). Noting that d ≤ l and that n ≤ 67k, we conclude that after guessing all the vertices in F μ , the problem can be solved in time O (27 l k).…”

Section: Is a Layer Decomposition Of G Then Clearly The Vertices In mentioning

confidence: 99%

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Parametric Duality and Kernelization: Lower Bounds and Upper Bounds on Kernel Size

Fernau²,

et al. 2007

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Abstract. Determining whether a parameterized problem is kernelizable and has a small kernel size has recently become one of the most interesting topics of research in the area of parameterized complexity and algorithms. Theoretically, it has been proved that a parameterized problem is kernelizable if and only if it is fixed-parameter tractable. Practically, applying a data reduction algorithm to reduce an instance of a parameterized problem to an equivalent smaller instance (i.e., a kernel) has led to very efficient algorithms and now goes hand-in-hand with the design of practical algorithms for solving N P-hard problems. Well-known examples of such parameterized problems include the vertex cover problem, which is kernelizable to a kernel of size bounded by 2k, and the planar dominating set problem, which is kernelizable to a kernel of size bounded by 335k. In this paper we develop new techniques to derive upper and lower bounds on the kernel size for certain parameterized problems. In terms of our lower bound results, we show, for example, that unless P = N P, planar vertex cover does not have a problem kernel of size smaller than 4k/3, and planar independent set and planar dominating set do not have kernels of size smaller than 2k. In terms of our upper bound results, we further reduce the upper bound on the kernel size for the planar dominating set problem to 67k, improving significantly the 335k previous upper bound given by Alber, Fellows, and Niedermeier [J. ACM, 51 (2004), pp. 363-384]. This latter result is obtained by introducing a new set of reduction and coloring rules, which allows the derivation of nice combinatorial properties in the kernelized graph leading to a tighter bound on the size of the kernel. The paper also shows how this improved upper bound yields a simple and competitive algorithm for the planar dominating set problem.Key words. parameterized algorithm, planar graph, dominating set, vertex cover, independent set, kernel AMS subject classifications. 05C85, 68Q17 DOI. 10.1137/050646354 1. Introduction. Many practical algorithms for N P-hard problems start by applying data reduction subroutines to the input instances of the problem. The hope is that after the data reduction phase the instance of the problem has shrunk to a moderate size. This makes the applicability of a second phase, such as a branchand-bound phase, to the resulting instance more feasible. Weihe showed in [41] how a practical preprocessing algorithm for a variation of the dominating set problem, called the red/blue dominating set problem, resulted in breaking up input instances of the problem into much smaller instances. Langston, and Shanbhag [2], in their implementation of algorithms for the vertex cover problem,

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Section: Is a Layer Decomposition Of G Then Clearly The Vertices In mentioning

confidence: 99%

Section: Is a Layer Decomposition Of G Then Clearly The Vertices In mentioning

confidence: 99%

Section: Is a Layer Decomposition Of G Then Clearly The Vertices In mentioning

confidence: 99%

Section: Is a Layer Decomposition Of G Then Clearly The Vertices In mentioning

confidence: 99%

See 2 more Smart Citations

Parametric Duality and Kernelization: Lower Bounds and Upper Bounds on Kernel Size

Fernau²,

et al. 2007

Self Cite

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show abstract

“…Even if a problem is shown to be FPT, the quest for improving the function f (k) goes on. For example, see Alber et al [2001Alber et al [ , 2002, Kanj and Perkovic [2002], and Fomin and Thilikos [2003] for recent attempts in improving the f (k) in the case of the Planar Dominating Set problem. The book by Downey and Fellows [1999] provides a good introduction to the topic of Parameterized Complexity.…”

Section: Introductionmentioning

confidence: 99%

Faster Fixed Parameter Tractable Algorithms for Undirected Feedback Vertex Set

Raman

Saurabh

Subramanian

2002

Algorithms and Computation

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Abstract. A feedback vertex set (fvs) of a graph is a set of vertices whose removal results in an acyclic graph. We show that if an undirected graph on n vertices with minimum degree at least 3 has a fvs on at most 1 3 n 1− vertices, then there is a cycle of length at most 6 (for ≥ 1/2, we can even improve this to just 6).Using this, we obtain a O(( 12 log k log log k + 6) k n ω ) algorithm for testing whether an undirected graph on n vertices has a fvs of size at most k. Here n ω is the complexity of the best matrix multiplication algorithm. The previous best parameterized algorithm for this problem took O((2k + 1) k n 2 ) time.We also investigate the fixed parameter complexity of weighted feedback vertex set problem in weighted undirected graphs.

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New upper bounds on the decomposability of planar graphs

Fomin

Thilikos

2005

Journal of Graph Theory

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DOI 10.1002/jgt.20121 Abstract: It is known that a planar graph on n vertices has branch-width/ tree-width bounded by ffiffiffi n p. In many algorithmic applications, it is useful to have a small bound on the constant . We give a proof of the best, so far, upper bound for the constant . In particular, for the case of tree-width, < 3:182 and for the case of branch-width, < 2:122. Our proof is based on the planar separation theorem of Alon, Seymour, and Thomas and some min–max theorems of Robertson and Seymour from the graph minors series. We also discuss some algorithmic consequences of this result

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Improved Parameterized Algorithms for Planar Dominating Set

Cited by 42 publications

References 8 publications

Parametric Duality and Kernelization: Lower Bounds and Upper Bounds on Kernel Size

Parametric Duality and Kernelization: Lower Bounds and Upper Bounds on Kernel Size

Faster Fixed Parameter Tractable Algorithms for Undirected Feedback Vertex Set

New upper bounds on the decomposability of planar graphs

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