FPT algorithms for Connected Feedback Vertex Set

Misra, Neeldhara; Philip, Geevarghese; Raman, Venkatesh; Saurabh, Saket; Sikdar, Somnath

doi:10.1007/s10878-011-9394-2

Cited by 33 publications

(22 citation statements)

References 21 publications

(29 reference statements)

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“…There are many papers dealing with vertex-partition problems on (di)graphs. Examples (from a long list) are [1,4,5,7,8,9,10,11,12,13,14,15,16,18,20,21,22,23]. Important examples for undirected graphs are bipartite graphs (those having has a 2-partition into two independent sets) and split graphs (those having a 2-partition into a clique and an independent set) [8].…”

Section: Introductionmentioning

confidence: 99%

Finding good 2-partitions of digraphs I. Hereditary properties

Bang‐Jensen

Havet

2016

Theoretical Computer Science

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We study the complexity of deciding whether a given digraph D has a vertex-partition into two disjoint subdigraphs with given structural properties. Let H and E denote following two sets of natural properties of digraphs: H ={acyclic, complete, arcless, oriented (no 2-cycle), semicomplete, symmetric, tournament} and E ={strongly connected, connected, minimum outdegree at least 1, minimum in-degree at least 1, minimum semi-degree at least 1, minimum degree at least 1, having an out-branching, having an in-branching}. In this paper, we determine the complexity of of deciding, for any fixed pair of positive integers k1, k2, whether a given digraph has a vertex partition into two digraphs D1, D2 such that |V (Di)| ≥ ki and Di has property Pi for i = 1, 2 when P1 ∈ H and P2 ∈ H ∪ E. We also classify the complexity of the same problems when restricted to strongly connected digraphs. The complexity of the problems when both P1 and P2 are in E is determined in the companion paper [2].

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Section: Introductionmentioning

confidence: 99%

Finding good 2-partitions of digraphs I. Hereditary properties

Bang‐Jensen

Havet

2016

Theoretical Computer Science

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“…Results in this paper have already been used to obtain kernelization lower bound in a few papers. These include recent results by Kratsch and Wahlström [23], Cygan et al [10] and Misra et al [24] who used UNIQUE COVERAGE, STEINER TREE and CONNECTED VERTEX COVER and CONNECTED VERTEX COVER respectively as a starting point to obtain polynomial parameter transformations. In another development Dell and van Melkebeek [11] have obtained a strengthening of a result in [17] and using that they are able to show concrete lower bounds on problems that do admit polynomial kernels.…”

Section: Conclusion Discussion and Further Workmentioning

confidence: 94%

Kernelization Lower Bounds Through Colors and IDs

Dom

Lokshtanov

Saurabh

2014

ACM Trans. Algorithms

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131

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In parameterized complexity each problem instance comes with a parameter k, and a parameterized problem is said to admit a polynomial kernel if there are polynomial time preprocessing rules that reduce the input instance to an instance with size polynomial in k. Many problems have been shown to admit polynomial kernels, but it is only recently that a framework for showing the non-existence of polynomial kernels for specific problems has been developed by Bodlaender et al. [6] and Fortnow and Santhanam [17]. With few exceptions , all known kernelization lower bounds result have been obtained by directly applying this framework. In this paper we show how to combine these results with combinatorial reductions which use colors and IDs in order to prove kernelization lower bounds for a variety of basic problems. Below we give a summary of our main results. All results are under the assumption that the polynomial hierarchy does not collapse to the third level. • We show that the STEINER TREE problem parameterized by the number of terminals and solution size k, and the CONNECTED VERTEX COVER and CAPACITATED VERTEX COVER problems do not admit a polynomial kernel. The two latter results are surprising because the closely related VERTEX COVER problem admits a kernel of with at most 2k vertices. • Alon and Gutner obtain a k poly(h) kernel for DOMINATING SET IN H-MINOR FREE GRAPHS parameterized by h = |H| and solution size k, and ask whether kernels of smaller size exist [3]. We partially resolve this question by showing that DOMINATING SET IN H-MINOR FREE GRAPHS does not admit a kernel with size polynomial in k + h. • Harnik and Naor obtain a "compression algorithm" for the SPARSE SUBSET SUM problem [21]. We show that their algorithm is essentially optimal by showing that the instances cannot be compressed further. • The HITTING SET and SET COVER problems are among the most studied problems in algorithmics. Both problems admit a kernel of size k O(d) when parameterized by solution size k and maximum set size d. We show that neither of them, along with the UNIQUE COVERAGE and BOUNDED RANK DISJOINT SETS problems, admits a polynomial kernel. The existence of polynomial kernels for several of the problems mentioned above were open problems explicitly stated in the literature [3, 4, 19, 20, 26]. Many of our results also rule out the existence of compression algorithms, a notion similar to kernelization defined by Harnik and Naor [21], for the problems in question.

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“…As IFVS is a more general problem than FVS, any improvement for IFVS will lead to an improvement for the FPT algorithm of FVS. We conclude with re-iterating an open problem of [23]: does there exist a kernel of size O(k 2 ), as it is the case for FVS [15,24]?…”

Section: Resultsmentioning

confidence: 99%

An Improved FPT Algorithm for Independent Feedback Vertex Set

Pilipczuk

2020

Theory Comput Syst

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We study the Independent Feedback Vertex Set problem -a variant of the classic Feedback Vertex Set problem where, given a graph G and an integer k, the problem is to decide whether there exists a vertex set S ⊆ V (G) such that G \ S is a forest and S is an independent set of size at most k. We present an O * ((1 + ϕ 2 ) k )-time FPT algorithm for this problem, where ϕ < 1.619 is the golden ratio, improving the previous fastest O * (4.1481 k )-time algorithm given by Agrawal et al. [2]. The exponential factor in our time complexity bound matches the fastest deterministic FPT algorithm for the classic Feedback Vertex Set problem.On the technical side, the main novelty is a refined measure of an input instance in a branching process, that allows for a simpler and more concise description and analysis of the algorithm. .pl. 1 The O * -notation suppresses factors that are polynomial in the input size. 2 Actually in the randomized FPT algorithm for FVS, the parameter is the treewidth of the graph. Since the treewidth of a yes-instance (G, k) to FVS is at most k + 1, the randomized algorithm for FVS runs in time O * (3 k ).

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FPT algorithms for Connected Feedback Vertex Set

Cited by 33 publications

References 21 publications

Finding good 2-partitions of digraphs I. Hereditary properties

Finding good 2-partitions of digraphs I. Hereditary properties

Kernelization Lower Bounds Through Colors and IDs

An Improved FPT Algorithm for Independent Feedback Vertex Set

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