Subset feedback vertex set on graphs of bounded independent set size

The weighted independent set problem on P 5 -free graphs has numerous applications, including data mining and dispatching in railways. The recognition of P 5 -free graphs is executed in polynomial time. Many problems, such as chromatic number and dominating set, are NP-hard in the class of P 5 -free graphs. The size of a minimum independent feedback vertex set that belongs to a P 5 -free graph with n vertices can be computed in O ( n 16 ) time. The unweighted problems, clique and clique cover, are NP-complete and the independent set is polynomial. In this work, the P 5 -free graphs using the weak decomposition are characterized, as is the dominating clique, and they are given an O ( n ( n + m ) ) recognition algorithm. Additionally, we calculate directly the clique number and the chromatic number; determine in O ( n ) time, the size of a minimum independent feedback vertex set; and determine in O ( n + m ) time the number of stability, the dominating number and the minimum clique cover.

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An Improved FPT Algorithm for Independent Feedback Vertex Set

Theory of Computing Systems ◽

10.1007/s00224-020-09973-w ◽

2020 ◽

Vol 64 (8) ◽

pp. 1317-1330

Author(s):

Shaohua Li ◽

Marcin Pilipczuk

Keyword(s):

Branching Process ◽

Golden Ratio ◽

Independent Set ◽

Time Algorithm ◽

Complexity Bound ◽

Feedback Vertex Set ◽

Exponential Factor ◽

Fpt Algorithm ◽

Vertex Set ◽

Input Instance

AbstractWe 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 $S\subseteq V(G)$ S ⊆ V ( G ) such that G ∖ S is a forest and S is an independent set of size at most k. We present an $\mathcal {O}^{\ast }((1+\varphi ^{2})^{k})$ O ∗ ( ( 1 + φ 2 ) k ) -time FPT algorithm for this problem, where φ < 1.619 is the golden ratio, improving the previous fastest $\mathcal {O}^{\ast }(4.1481^{k})$ O ∗ ( 4.148 1 k ) -time algorithm given by Agrawal et al. (2016). 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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Parameterized complexity of fair feedback vertex set problem

Theoretical Computer Science ◽

10.1016/j.tcs.2021.03.008 ◽

2021 ◽

Vol 867 ◽

pp. 1-12

Author(s):

Lawqueen Kanesh ◽

Soumen Maity ◽

Komal Muluk ◽

Saket Saurabh

Keyword(s):

Parameterized Complexity ◽

Feedback Vertex Set ◽

Vertex Set ◽

Feedback Vertex Set Problem

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Parameterized Algorithms for Generalizations of Directed Feedback Vertex Set

Lecture Notes in Computer Science - Algorithms and Complexity ◽

10.1007/978-3-030-17402-6_21 ◽

2019 ◽

pp. 249-261

Author(s):

Alexander Göke ◽

Dániel Marx ◽

Matthias Mnich

Keyword(s):

Parameterized Algorithms ◽

Feedback Vertex Set ◽

Vertex Set ◽

Directed Feedback Vertex Set

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On Structural Parameterizations of the Edge Disjoint Paths Problem

Algorithmica ◽

10.1007/s00453-020-00795-3 ◽

2021 ◽

Author(s):

Robert Ganian ◽

Sebastian Ordyniak ◽

M. S. Ramanujan

Keyword(s):

Polynomial Time ◽

Disjoint Paths ◽

Structural Parameter ◽

Input Graph ◽

Feedback Vertex Set ◽

Fpt Algorithms ◽

Edge Disjoint ◽

Fpt Algorithm ◽

Vertex Set ◽

Terminal Pair

AbstractIn this paper we revisit the classical edge disjoint paths (EDP) problem, where one is given an undirected graph G and a set of terminal pairs P and asks whether G contains a set of pairwise edge-disjoint paths connecting every terminal pair in P. Our focus lies on structural parameterizations for the problem that allow for efficient (polynomial-time or FPT) algorithms. As our first result, we answer an open question stated in Fleszar et al. (Proceedings of the ESA, 2016), by showing that the problem can be solved in polynomial time if the input graph has a feedback vertex set of size one. We also show that EDP parameterized by the treewidth and the maximum degree of the input graph is fixed-parameter tractable. Having developed two novel algorithms for EDP using structural restrictions on the input graph, we then turn our attention towards the augmented graph, i.e., the graph obtained from the input graph after adding one edge between every terminal pair. In constrast to the input graph, where EDP is known to remain -hard even for treewidth two, a result by Zhou et al. (Algorithmica 26(1):3--30, 2000) shows that EDP can be solved in non-uniform polynomial time if the augmented graph has constant treewidth; we note that the possible improvement of this result to an FPT-algorithm has remained open since then. We show that this is highly unlikely by establishing the [1]-hardness of the problem parameterized by the treewidth (and even feedback vertex set) of the augmented graph. Finally, we develop an FPT-algorithm for EDP by exploiting a novel structural parameter of the augmented graph.

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