A fixed-parameter algorithm for the directed feedback vertex set problem

Chen, Jianer; Liu, Yang; Lu, Songjian; O’Sullivan, Barry; Razgon, Igor

doi:10.1145/1411509.1411511

Cited by 232 publications

(179 citation statements)

References 26 publications

(20 reference statements)

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“…The fixed-parameter tractability of the directed version remained an open problem until Chen et al [8] presented a para-P-algorithm in 2008. So far, these apparently different complexities could only be "felt" by looking at the complexity of the proofs.…”

Section: The Complexity Of the Feedback Vertex Set Problemmentioning

confidence: 99%

On the Space and Circuit Complexity of Parameterized Problems: Classes and Completeness

2014

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The parameterized complexity of a problem is generally considered "settled" once it has been shown to be fixed-parameter tractable or to be complete for a class in a parameterized hierarchy such as the weft hierarchy. Several natural parameterized problems have, however, resisted such a classification. In the present paper we argue that, in some cases, this is due to the fact that the parameterized complexity of these problems can be better understood in terms of their parameterized space or parameterized circuit complexity. This includes well-studied, natural problems like the feedback vertex set problem, the associative generability problem, or the longest common subsequence problem. We show that these problems lie in and may even be complete for different parameterized space classes, leading to new insights into the problems' complexity. The classes we study are defined in terms of different forms of bounded nondeterminism and simultaneous time-space bounds.

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Section: The Complexity Of the Feedback Vertex Set Problemmentioning

confidence: 99%

On the Space and Circuit Complexity of Parameterized Problems: Classes and Completeness

2014

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“…Similar to the pushing lemma for Skew Edge Multicut [4], we propose a pushing lemma for the multi-budgeted variant.…”

Section: Multi-budgeted Skew Edge Multicutmentioning

confidence: 99%

“…The notion of important separators, introduced by Marx [23], turned out to be fundamental for a number of graph separation problems such as Multiway Cut [23], Directed Feedback Vertex Set [4], or Almost 2-CNF SAT [28]. Further work, concerning mostly undirected graphs, resulted in a wide range of involved algorithmic techniques: applications of matroid techniques [19,20], shadow removal [8,26], randomized contractions [5], LP-guided branching [10,15,16,17], and treewidth reduction [25], among others.…”

Section: Introductionmentioning

confidence: 99%

Multi-budgeted Directed Cuts

Kratsch

Marx

et al. 2019

Algorithmica

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In this paper, we study multi-budgeted variants of the classic minimum cut problem and graph separation problems that turned out to be important in parameterized complexity: Skew Multicut and Directed Feedback Arc Set. In our generalization, we assign colors 1, 2, ..., to some edges and give separate budgets k1, k2, ..., k for colors 1, 2, ..., . For every color i ∈ {1, ..., }, let Ei be the set of edges of color i. The solution C for the multi-budgeted variant of a graph separation problem not only needs to satisfy the usual separation requirements (i.e., be a cut, a skew multicut, or a directed feedback arc set, respectively), but also needs to satisfy that |C ∩ Ei| ≤ ki for every i ∈ {1, ..., }.Contrary to the classic minimum cut problem, the multi-budgeted variant turns out to be NP-hard even for = 2. We propose FPT algorithms parameterized by k = k1 + ... + k for all three problems. To this end, we develop a branching procedure for the multi-budgeted minimum cut problem that measures the progress of the algorithm not by reducing k as usual, by but elevating the capacity of some edges and thus increasing the size of maximum source-to-sink flow. Using the fact that a similar strategy is used to enumerate all important separators of a given size, we merge this process with the flow-guided branching and show an FPT bound on the number of (appropriately defined) important multi-budgeted separators. This allows us to extend our algorithm to the Skew Multicut and Directed Feedback Arc Set problems.Furthermore, we show connections of the multi-budgeted variants with weighted variants of the directed cut problems and the Chain -SAT problem, whose parameterized complexity remains an open problem. We show that these problems admit a bounded-in-parameter number of "maximally pushed" solutions (in a similar spirit as important separators are maximally pushed), giving somewhat weak evidence towards their tractability.

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“…The undirected version was shown to be fixed-parameter tractable in 1993 by Bodlaender [2]. The fixed-parameter tractability of the directed version remained an open problem until Chen et al [6] presented a para-P-algorithm in 2008. So far, these apparently different complexities could only be felt by looking at the complexity of the proofs.…”

Section: Feedback Vertex Set Problemsmentioning

confidence: 99%

On the Space Complexity of Parameterized Problems

Elberfeld

Stockhusen

Tantau

2012

Parameterized and Exact Computation

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Parameterized complexity theory measures the complexity of computational problems predominantly in terms of their parameterized time complexity. The purpose of the present paper is to demonstrate that the study of parameterized space complexity can give new insights into the complexity of well-studied parameterized problems like the feedback vertex set problem. We show that the undirected and the directed feedback vertex set problems have different parameterized space complexities, unless L = NL. For a number of further natural parameterized problems, including the longest common subsequence problem, the acceptance problem for multi-head automata, and the associative generability problem we show that they lie in or are complete for different parameterized space classes. Our results explain why previous attempts at proving completeness of different problems for parameterized time classes have failed.

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A fixed-parameter algorithm for the directed feedback vertex set problem

Cited by 232 publications

References 26 publications

On the Space and Circuit Complexity of Parameterized Problems: Classes and Completeness

On the Space and Circuit Complexity of Parameterized Problems: Classes and Completeness

Multi-budgeted Directed Cuts

On the Space Complexity of Parameterized Problems

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