Parameterized approximation of dominating set problems

Downey, Rodney G.; Fellows, Michael R.; McCartin, Catherine; Rosamond, Frances A.

doi:10.1016/j.ipl.2008.09.017

Cited by 43 publications

(22 citation statements)

References 9 publications

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“…Recently, it has been proposed that the notion of approximability can be investigated in the framework of fixed-parameter tractability, and various models have been suggested (see, e.g., [9,11,16]. These models seek, for example, an FPT-algorithm which on input k either delivers "a no size k dominating set" or produces one of size 2k.…”

Section: Related Workmentioning

confidence: 99%

“…The algorithms developed for such problems yield a solution of value g(k) for a problem parameterized by k, where k is the solution size (see, e.g., [20,16,18]). …”

Section: Related Workmentioning

confidence: 99%

“…A similar approach was developed by Cygan et al [12]. Moderately exponential approximation has been investigated by [9,11,16], though with objectives oriented towards development of fixed-parameter algorithms.…”

Section: Related Workmentioning

confidence: 99%

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Parameterized Approximation via Fidelity Preserving Transformations

Fellows

Kulik

Rosamond

et al. 2012

Automata, Languages, and Programming

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Abstract.We motivate and describe a new parameterized approximation paradigm which studies the interaction between performance ratio and running time for any parameterization of a given optimization problem. As a key tool, we introduce the concept of α-shrinking transformation, for α ≥ 1. Applying such transformation to a parameterized problem instance decreases the parameter value, while preserving approximation ratio of α (or α-fidelity). For example, it is well-known that Vertex Cover cannot be approximated within any constant factor better than 2 [22] (under usual assumptions). Our parameterized α-approximation algorithm for k-Vertex Cover, parameterized by the solution size, has a running time of 1.273(2−α)k , where the running time of the best FPT algorithm is 1.273 k [10]. Our algorithms define a continuous tradeoff between running times and approximation ratios, allowing practitioners to appropriately allocate computational resources. Moving even beyond the performance ratio, we call for a new type of approximative kernelization race. Our α-shrinking transformations can be used to obtain kernels which are smaller than the best known for a given problem. For the Vertex Cover problem we obtain a kernel size of 2(2 − α)k. The smaller "α-fidelity" kernels allow us to solve exactly problem instances more efficiently, while obtaining an approximate solution for the original instance. We show that such transformations exist for several fundamental problems, including Vertex Cover, d-Hitting Set, Connected Vertex Cover and Steiner Tree. We note that most of our algorithms are easy to implement and are therefore practical in use.

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Section: Related Workmentioning

confidence: 99%

“…The algorithms developed for such problems yield a solution of value g(k) for a problem parameterized by k, where k is the solution size (see, e.g., [20,16,18]). …”

Section: Related Workmentioning

confidence: 99%

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Parameterized Approximation via Fidelity Preserving Transformations

Fellows

Kulik

Rosamond

et al. 2012

Automata, Languages, and Programming

Self Cite

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“…We may also be able to use this technique to handle W[1]-hard problems which unlikely have fixed-parameter tractable algorithms. The interested reader can be referred to [4,11,19] for more about this issue. Let the parameter k be the size of the solution to our problem.…”

Section: Introductionmentioning

confidence: 99%

“…In parameterized approximation, we wish to design algorithms with running time f (k)|I| O(1) that either find an approximate solution of size g(k) or report that there is no solution of size k. Clearly, any fixed-parameter tractable problem allows parameterized approximation algorithms for any computable function g. However, this may not hold for W [1]-hard problems. For example, the dominating set problem (find a set S of k vertices in graph G = (E, V ) such that each vertex in V − S is adjacent to at least one vertex in S) does not allow parameterized approximation algorithms for g(k) of the form k + c with fixed constant c [11]. For edge dominating set, we are interested in designing parameterized approximation algorithms, which produce edge dominating sets of size at most (1 + ε)k (or assert that there is no solution of size k) in f (k, ε)|I| O (1) time for some computable function f .…”

Section: Introductionmentioning

confidence: 99%

New Results on Polynomial Inapproximabilityand Fixed Parameter Approximability of Edge Dominating Set

et al. 2014

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Abstract. An edge dominating set in a graph G = (V, E) is a subset S of edges such that each edge in E − S is adjacent to at least one edge in S. The edge dominating set problem, to find an edge dominating set of minimum size, is a basic and important NP-hard problem that has been extensively studied in approximation algorithms and parameterized complexity. In this paper, we present improved hardness results and parameterized approximation algorithms for edge dominating set. More precisely, we first show that it is NP-hard to approximate edge dominating set in polynomial time within a factor better than 1.18. Next, we give a parameterized approximation schema (with respect to the standard parameter) for the problem and, finally, we develop an O * (1.821 τ )-time exact algorithm where τ is the size of a minimum vertex cover of G.

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Finding hidden hubs and dominating sets in sparse graphs by randomized neighborhood queries

Damaschke

2010

Networks

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Suppose that we have (i) a graph with an unknown edge set and (ii) access to an oracle that returns all neighbors of a probed vertex. We want to find the hubs, that is, vertices with degree above a given threshold. An almost obvious two-stage randomized strategy is to query randomly sampled vertices and then their neighbors. We prove that this strategy achieves, in certain classes of sparse graphs, an asymptotically optimal query number among all possible querying strategies, including adaptive strategies. A detail of importance for an optimal constant factor is the number of probes in the neighborhood of a hub candidate. We show that 2 is in general the best choice in the graphs of interest. In a similar way we analyze, in the same model, the problem of finding small, almost dominating sets. The problems and graph classes are motivated mainly by the elucidation of interaction networks in molecular biology.

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Parameterized approximation of dominating set problems

Cited by 43 publications

References 9 publications

Parameterized Approximation via Fidelity Preserving Transformations

Parameterized Approximation via Fidelity Preserving Transformations

New Results on Polynomial Inapproximabilityand Fixed Parameter Approximability of Edge Dominating Set

Finding hidden hubs and dominating sets in sparse graphs by randomized neighborhood queries

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