Optimization of Pearl's method of conditioning and greedy-like approximation algorithms for the vertex feedback set problem

Becker, Ann; Geiger, Dan

doi:10.1016/0004-3702(95)00004-6

Cited by 110 publications

(108 citation statements)

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“…So, suppose k ≥ 1. The first step of the kernelization algorithm is to run the approximation algorithm of Bafna et al [3] or that of Becker and Geiger [5]. These algorithms have a performance ratio of 2.…”

Section: Initialization Phasementioning

confidence: 99%

“…In such a way, the algorithm could be used as a preprocessing heuristic. We first start with setting k to the size of the feedback vertex set A, returned by the approximation algorithm of Bafna et al [3] or the algorithm of Becker and Geiger [5].…”

Section: Without An Initial Value Of Kmentioning

confidence: 99%

“…If we use, instead of, or in addition to, the algorithm of Bafna et al [3] or Becker and Geiger [5] some other algorithms that finds close to optimal feedback vertex sets, and this algorithm finds a feedback vertex set that is smaller, then the guaranteed bound on the size of the kernel is also smaller. For instance, we could try to improve upon a solution found by the algorithms of [3,5] with a local search algorithm.…”

Section: With a Different Algorithm To Find The Initial Feedback Vertmentioning

confidence: 99%

“…The approximation algorithms of Bafna et al [3] and of Becker and Geiger [5] also can solve weighted versions of the feedback vertex set problem. This can also be of help for preprocessing and kernelization of the unweighted problem.…”

Section: Strongly Forced Verticesmentioning

confidence: 99%

“…In this paper, we improve on the size of this kernel, and show that FEEDBACK VERTEX SET has a kernel with O(k 3 ) vertices. The kernelization algorithm uses the 2-approximation algorithm for FEEDBACK VERTEX SET of [3] or [5] as a first step, and then uses a set of relatively simple reduction rules. A combinatorial proof shows that if no rule can be applied, then the graph has O(k 3 ) vertices, where k is the parameter in the reduced instance.…”

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confidence: 99%

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A Cubic Kernel for Feedback Vertex Set and Loop Cutset

Bodlaender

Dijk

2009

Theory Comput Syst

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The FEEDBACK VERTEX SET problem on unweighted, undirected graphs is considered. Improving upon a result by Burrage et al. (Proceedings 2nd International Workshop on Parameterized and Exact Computation, pp. 192-202, 2006), we show that this problem has a kernel with O(k 3 ) vertices, i.e., there is a polynomial time algorithm, that given a graph G and an integer k, finds a graph G with O(k 3 ) vertices and integer k ≤ k, such that G has a feedback vertex set of size at most k, if and only if G has a feedback vertex set of size at most k . Moreover, the algorithm can be made constructive: if the reduced instance G has a feedback vertex set of size k , then we can easily transform a minimum size feedback vertex set of G into a minimum size feedback vertex set of G. This kernelization algorithm can be used as the first step of an FPT algorithm for FEEDBACK VERTEX SET, but also as a preprocessing heuristic for FEEDBACK VERTEX SET.We also show that the related LOOP CUTSET problem also has a kernel of cubic size. The kernelization algorithms are experimentally evaluated, and we report on these experiments.

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Section: Initialization Phasementioning

confidence: 99%

Section: Without An Initial Value Of Kmentioning

confidence: 99%

Section: With a Different Algorithm To Find The Initial Feedback Vertmentioning

confidence: 99%

Section: Strongly Forced Verticesmentioning

confidence: 99%

mentioning

confidence: 99%

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A Cubic Kernel for Feedback Vertex Set and Loop Cutset

Bodlaender

Dijk

2009

Theory Comput Syst

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Minimum k‐cores and the k‐core polytope

Mikesell

Hicks

2021

Networks

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The minimum k-core problem asks for the smallest induced subgraph of minimum degree k. It has been shown that this problem is NP-hard, and thus sophisticated techniques are required to obtain good solutions and approximations. In this article, the minimum k-core problem is modeled as a binary integer program and relaxed as a linear program. Since the relaxation may yield a non-integral solution, a branch-and-cut framework is used to find an integral optimal solution. It is shown that the edge and cycle transversals of the graph give valid inequalities for the convex hull of the k-core polytope-which can be further generalized to a family of 𝓁-core transversals. Further, a heuristic for the transversal of the minimal 𝓁-cores is given with its associated valid inequality. Additionally, improved valid inequalities are generated using bounds involving the girth of the graph. Multiple heuristics are explored for finding initial bounds for the branching process utilizing the degree distribution of the graph. Finally, numerical results are given comparing the branch-and-bound, branch-and-cut, and heuristic techniques.

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Biquadratic assignment problem; Graph coloring; Graph planarization; Greedy randomized adaptive search procedures; Quadratic assignment problem; Quadratic semi-assignment problem; Three-index assignment problemBiquadratic assignment problem; Graph coloring; Graph planarization; Greedy randomized adaptive search procedures; Linear ordering problem; Quadratic assignment problem; Quadratic semi-assignment problem; Three-index assignment problemFEEDBACK SET PROBLEMS

Festa¹,

Via²,

Pardalos³

et al. 2001

Encyclopedia of Optimization

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Optimization of Pearl's method of conditioning and greedy-like approximation algorithms for the vertex feedback set problem

Cited by 110 publications

References 10 publications

A Cubic Kernel for Feedback Vertex Set and Loop Cutset

A Cubic Kernel for Feedback Vertex Set and Loop Cutset

Minimum k‐cores and the k‐core polytope

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