An Approximation Algorithm for Feedback Vertex Sets in Tournaments

Cai, Mao-cheng; Deng, Xiaotie; Zang, Wenan

doi:10.1137/s0097539798338163

Cited by 60 publications

(64 citation statements)

References 16 publications

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“…We show that the equivalence between the paradigms continues to hold. We demonstrate the use of the extended frameworks on several algorithms: a 2.5-approximation algorithm for feedback vertex set in tournaments [16]; a 2-approximation algorithm for a noncovering problem called minimum 2-satisfiability [29,9]; and a 3-approximation algorithm for a bandwidth trading problem [15]. We show that the equivalence continues to hold in the maximization case.…”

Section: Our Resultsmentioning

confidence: 98%

On the Equivalence between the Primal-Dual Schema and the Local Ratio Technique

Bar-Yehuda¹,

Rawitz²

2005

SIAM J. Discrete Math.

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Abstract. We discuss two approximation paradigms that were used to construct many approximation algorithms during the last two decades, the primal-dual schema and the local ratio technique. Recently, primal-dual algorithms were devised by first constructing a local ratio algorithm and then transforming it into a primal-dual algorithm. This was done in the case of the 2-approximation algorithms for the feedback vertex set problem and in the case of the first primal-dual algorithms for maximization problems. Subsequently, the nature of the connection between the two paradigms was posed as an open question by Williamson [Math. Program., 91 (2002), pp. 447-478]. In this paper we answer this question by showing that the two paradigms are equivalent.

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Section: Our Resultsmentioning

confidence: 98%

On the Equivalence between the Primal-Dual Schema and the Local Ratio Technique

Bar-Yehuda¹,

Rawitz²

2005

SIAM J. Discrete Math.

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“…This yields a branching vector of (7,7,5,3,5,7,4,3) This yields a branching vector of (7,7,7,3,1) This yields a branching vector of (5, 7, 7, 7, 5, 7, 1) that solves to 1.5793. Henceforth, we can assume that 2 , 3 have no common predecessor except possibly 1 .…”

Section: Case 232mentioning

confidence: 99%

“…In other words, a tournament is a digraph with exactly one arc between any two of its vertices. Various approaches have been suggested to solve the MINIMUM FVS problem on tournaments, including approximation algorithms [3,11], fixed-parameter algorithms [4,9] as well asalgorithm [6,13]. The running time of this approach is within a polynomial factor of the number ( ) of minimal FVS in .…”

Section: Introductionmentioning

confidence: 99%

Improved bounds for minimal feedback vertex sets in tournaments

Mnich

Teutrine

2017

Journal of Graph Theory

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“…This structure can often be exploited to obtain algorithms with better performance than the corresponding 3-Hitting Set algorithm. In particular, for FVST, Cai et al [8] gave a factor 2.5 approximation algorithm. This has recently been improved to 7/3 by Mnich et al [29].…”

Section: Introductionmentioning

confidence: 99%

Subquadratic Kernels for Implicit 3-Hitting Set and 3-Set Packing Problems

Lokshtanov¹,

Saurabh²,

Thomassé³

et al. 2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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We consider four well-studied NP-complete packing/covering problems on graphs: Feedback Vertex Set in Tournaments (FVST), Cluster Vertex Deletion (CVD), Triangle Packing in Tournaments (TPT) and Induced P 3 -Packing. For these four problems kernels with O(k 2 ) vertices have been known for a long time. In fact, such kernels can be obtained by interpreting these problems as finding either a packing of k pairwise disjoint sets of size 3 (3-Set Packing) or a hitting set of size at most k for a family of sets of size at most 3 (3-Hitting Set). In this paper, we give the first kernels for FVST, CVD, TPT and Induced P 3 -Packing with a subquadratic number of vertices. Specifically, we obtain the following results.• FVST admits a kernel with O(k 3 2 ) vertices.• CVD admits a kernel with O(k 5 3 ) vertices.• TPT admits a kernel with O(k 3 2 ) vertices.• Induced P 3 -Packing admits a kernel with O(k Our results resolve an open problem from WorKer 2010 on the existence of kernels with O(k 2− ) vertices for FVST and CVD. All of our results are based on novel uses of old and new "expansion lemmas", and a weak form of crown decomposition where (i) almost all of the head is used by the solution (as opposed to all), (ii) almost none of the crown is used by the solution (as opposed to none), and (iii) if H is removed from G, then there is almost no interaction between the head and the rest (as opposed to no interaction at all).

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An Approximation Algorithm for Feedback Vertex Sets in Tournaments

Cited by 60 publications

References 16 publications

On the Equivalence between the Primal-Dual Schema and the Local Ratio Technique

On the Equivalence between the Primal-Dual Schema and the Local Ratio Technique

Improved bounds for minimal feedback vertex sets in tournaments

Subquadratic Kernels for Implicit 3-Hitting Set and 3-Set Packing Problems

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