Improved Algorithms and Combinatorial Bounds for Independent Feedback Vertex Set

Agrawal, Akanksha; Gupta, Sushmita; Saurabh, Saket; Sharma, Roohani

doi:10.4230/lipics.ipec.2016.2

Cited by 4 publications

(13 citation statements)

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“…[26]) admit elegant duality theorems such as the Erdös-Pósa property; understanding their use in designing algorithms can be instrumental to solve many problems diferent from FVS faster. The popularity of FVS also led to work on a broad spectrum of its variations such as Subset, Group, Connected, Simultaneous, or Independent FVS (see for example [2] and the references therein).…”

Section: Introductionmentioning

confidence: 99%

Detecting Feedback Vertex Sets of Size k in O ^⋆ (2.7 k ) Time

Nederlof

2022

ACM Trans. Algorithms

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In the Feedback Vertex Set problem, one is given an undirected graph G and an integer k , and one needs to determine whether there exists a set of k vertices that intersects all cycles of G (a so-called feedback vertex set). Feedback Vertex Set is one of the most central problems in parameterized complexity: It served as an excellent test bed for many important algorithmic techniques in the field such as Iterative Compression [Guo et al. (JCSS’06)], Randomized Branching [Becker et al. (J. Artif. Intell. Res’00)] and Cut&Count [Cygan et al. (FOCS’11)]. In particular, there has been a long race for the smallest dependence f ( k ) in run times of the type O ⋆ ( f ( k )), where the O ⋆ notation omits factors polynomial in n . This race seemed to have reached a conclusion in 2011, when a randomized O ⋆ (3 k ) time algorithm based on Cut&Count was introduced. In this work, we show the contrary and give a O ⋆ (2.7 k ) time randomized algorithm. Our algorithm combines all mentioned techniques with substantial new ideas: First, we show that, given a feedback vertex set of size k of bounded average degree, a tree decomposition of width (1 − Ω (1)) k can be found in polynomial time. Second, we give a randomized branching strategy inspired by the one from [Becker et al. (J. Artif. Intell. Res’00)] to reduce to the aforementioned bounded average degree setting. Third, we obtain significant run time improvements by employing fast matrix multiplication.

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Section: Introductionmentioning

confidence: 99%

Detecting Feedback Vertex Sets of Size k in O ^⋆ (2.7 k ) Time

Nederlof

2022

ACM Trans. Algorithms

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“…In this paper, we focus on the parameterized version of the Independent Feedback Vertex Set problem (IFVS), which is to decide if there exists a feedback vertex set S of size at most k such that no two vertices of S are adjacent in G. Misra and an O(k 3 ) kernel for IFVS [22]. Agrawal et al presented an improved FPT algorithm running in time O * (4.1481 k ) for IFVS [2]. In this paper, we propose a faster FPT algorithm.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…On the technical side, we follow the standard approach of iterative compression as in Agrawal et al [2] to reduce to a "disjoint" version of the problem. Here, our approach diverges from the one of Agrawal et al [2]. We follow a modified measure for the subsequent branching process, somewhat inspired by the work of Kociumaka and Pilipczuk [18].…”

Section: Introductionmentioning

confidence: 99%

“…

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.

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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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Improved FPT Algorithms for Deletion to Forest-Like Structures

Gowda,

Lonkar,

Panolan

et al. 2024

Algorithmica

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The Feedback Vertex Set problem is undoubtedly one of the most well-studied problems in Parameterized Complexity. In this problem, given an undirected graph G and a non-negative integer k, the objective is to test whether there exists a subset S ⊆ V (G) of size at most k such that G − S is a forest. After a long line of improvement, recently, Li and Nederlof [SODA, 2020] designed a randomized algorithm for the problem running in time O (2.7 k ) 1 . In the Parameterized Complexity literature, several problems around Feedback Vertex Set have been studied. Some of these include Independent Feedback Vertex Set (where the set S should be an independent set in G), Almost Forest Deletion and Pseudoforest Deletion. In Pseudoforest Deletion, each connected component in G − S has at most one cycle in it. However, in Almost Forest Deletion, the input is a graph G and non-negative integers k, ∈ N, and the objective is to test whether there exists a vertex subset S of size at most k, such that G − S is edges away from a forest. In this paper, using the methodology of Li and Nederlof [SODA, 2020], we obtain the current fastest algorithms for all these problems. In particular we obtain following randomized algorithms. 1. Independent Feedback Vertex Set can be solved in time O (2.7 k ). 2. Pseudo Forest Deletion can be solved in time O (2.85 k ). 3. Almost Forest Deletion can be solved in O (min{2.85 k • 8.54 , 2.7 k • 36.61 , 3 k • 1.78 }).

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Improved Algorithms and Combinatorial Bounds for Independent Feedback Vertex Set

Cited by 4 publications

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Detecting Feedback Vertex Sets of Size k in O ^⋆ (2.7 k ) Time

Detecting Feedback Vertex Sets of Size k in O ^⋆ (2.7 k ) Time

An Improved FPT Algorithm for Independent Feedback Vertex Set

Improved FPT Algorithms for Deletion to Forest-Like Structures

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Improved Algorithms and Combinatorial Bounds for Independent Feedback Vertex Set

Cited by 4 publications

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Detecting Feedback Vertex Sets of Size k in O ⋆ (2.7 k ) Time

Detecting Feedback Vertex Sets of Size k in O ⋆ (2.7 k ) Time

An Improved FPT Algorithm for Independent Feedback Vertex Set

Improved FPT Algorithms for Deletion to Forest-Like Structures

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Detecting Feedback Vertex Sets of Size k in O ^⋆ (2.7 k ) Time

Detecting Feedback Vertex Sets of Size k in O ^⋆ (2.7 k ) Time