A 2 + ɛ approximation algorithm for the k-MST problem

Arora, Sanjeev; Karakostas, George

doi:10.1007/s10107-005-0693-1

Cited by 47 publications

(68 citation statements)

References 13 publications

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“…Partial cover problems have been investigated extensively and are well understood in the context of polynomial time approximation [2,4,3,5,16,18] and parameterized complexity [1,4,24,25,23,27]. In this paper we study partial cover problems defined on graphs namely Partial Vertex Cover and Partial r-Dominating Set from the view point of parameterized algorithms.…”

Section: Introductionmentioning

confidence: 99%

Subexponential algorithms for partial cover problems

Fomin

Lokshtanov

Raman

et al. 2011

Information Processing Letters

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Partial Cover problems are optimization versions of fundamental and well studied problems like Vertex Cover and Dominating Set. Here one is interested in covering (or dominating) the maximum number of edges (or vertices) using a given number (k) of vertices, rather than covering all edges (or vertices). In general graphs, these problems are hard for parameterized complexity classes when parameterized by k. It was recently shown by Amini et. al. [FSTTCS 08 ] that Partial Vertex Cover and Partial Dominating Set are fixed parameter tractable on large classes of sparse graphs, namely H-minor free graphs, which include planar graphs and graphs of bounded genus. In particular, it was shown that on planar graphs both problems can be solved in time 2 O(k) n O(1) .During the last decade there has been an extensive study on parameterized subexponential algorithms. In particular, it was shown that the classical Vertex Cover and Dominating Set problems can be solved in subexponential time on H-minor free graphs. The techniques developed to obtain subexponential algorithms for classical problems do not apply to partial cover problems. It was left as an open problem by Amini et al. [FSTTCS 08 ] whether there is a subexponential algorithm for Partial Vertex Cover and Partial Dominating Set. In this paper, we answer the question affirmatively by solving both problems in time 2 O( √ k) n O(1) not only on planar graphs but also on much larger classes of graphs, namely, apex-minor free graphs. Compared to previously known algorithms for these problems our algorithms are significantly faster and simpler.

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Section: Introductionmentioning

confidence: 99%

Subexponential algorithms for partial cover problems

Fomin

Lokshtanov

Raman

et al. 2011

Information Processing Letters

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“…The problem also received a lot of attention in the * Technical University of Dortmund; {markus.chimani, maria.kandyba, petra.mutzel}@cs.uni-dortmund.de † Supported by the German Research Foundation (DFG) through the Collaborative Research Center "Computational Intelligence" (SFB 531) ‡ University of Vienna; ivana.ljubic@univie.ac.at § Supported by the Hertha-Firnberg Fellowship of the Austrian Science Foundation (FWF) approximation algorithm community [1,3,17,18]: a central idea thereby is the primal-dual scheme, based on integer linear programs (ILPs), which was proposed by Goemans and Williamson [19] for the prizecollecting Steiner tree problem. An exact approach was presented by Fischetti et al [15], by formulating an ILP based on general subtour elimination constraints (Gsec).…”

Section: Introductionmentioning

confidence: 99%

Obtaining Optimalk-Cardinality Trees Fast

Chimani

Kandyba

Ljubić

et al. 2008

2008 Proceedings of the Tenth Workshop on Algorithm Engineering and Experiments (ALENEX)

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Given an undirected graph G = (V, E) with edge weights and a positive integer number k, the k-Cardinality Tree problem consists of finding a subtree T of G with exactly k edges and the minimum possible weight. Many algorithms have been proposed to solve this NP-hard problem, resulting in mainly heuristic and metaheuristic approaches.In this paper we present an exact ILP-based algorithm using directed cuts. We mathematically compare the strength of our formulation to the previously known ILP formulations of this problem, and give an extensive study on the algorithm's practical performance compared to the state-of-the-art metaheuristics.In contrast to the widespread assumption that such a problem cannot be efficiently tackled by exact algorithms for medium and large graphs (between 200 and 5000 nodes), our results show that our algorithm not only has the advantage of proving the optimality of the computed solution, but also often outperforms the metaheuristic approaches in terms of running time.

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“…However, in practice, w i , the weight of primary vertex i in WAG, is a continuous value, so even a large scaling factor Δ cannot guarantee Δ * w i is an integer for all i. Furthermore, the time complexity of state-of-the-art k-MST approximation algorithms [11], [6], [1] are polynomial to the number of vertices in G , which is proportional to Δ. So it is advisable to use a medium Δ and let the number of subsidiaries p i rounded as Δ * w i -1.…”

Section: Theorem 33mentioning

confidence: 99%

“…2PASS works under the granularity metric, which is the predominant privacy definition in clientserver environments, for its simplicity and user-friendliness. 1 If a single cell still does not meet the privacy requirement, that is, the area of this cell is smaller than the threshold, then the cloaked region must span more cells, which means the user must request more objects. To minimize the number of non-result objects, we reduce this problem to the k-minimum spanning tree (k-MST) problem and provide an efficient and yet close-to-optimal algorithm to select objects to request.…”

Section: Introductionmentioning

confidence: 99%

2PASS: Bandwidth-Optimized Location Cloaking for Anonymous Location-Based Services

2010

IEEE Trans. Parallel Distrib. Syst.

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A 2 + ɛ approximation algorithm for the k-MST problem

Cited by 47 publications

References 13 publications

Subexponential algorithms for partial cover problems

Subexponential algorithms for partial cover problems

Obtaining Optimalk-Cardinality Trees Fast

2PASS: Bandwidth-Optimized Location Cloaking for Anonymous Location-Based Services

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