scholarly journals An Improved FPT Algorithm for Independent Feedback Vertex Set

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.

scholarly journals On Structural Parameterizations of the Edge Disjoint Paths Problem

Algorithmica ◽  
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.

An O(2 O(k) n 3) FPT Algorithm for the Undirected Feedback Vertex Set Problem

2005 ◽  
pp. 859-869 ◽  
Author(s):  
Frank Dehne ◽  
Michael Fellows ◽  
Michael A. Langston ◽  
Frances Rosamond ◽  
Kim Stevens
Keyword(s):  
Feedback Vertex Set ◽  
Fpt Algorithm ◽  
Vertex Set ◽  
Feedback Vertex Set Problem

An O(2O(k)n3) FPT Algorithm for the Undirected Feedback Vertex Set Problem

2007 ◽  
Vol 41 (3) ◽  
pp. 479-492 ◽  
Author(s):  
Frank Dehne ◽  
Michael Fellows ◽  
Michael Langston ◽  
Frances Rosamond ◽  
Kim Stevens
Keyword(s):  
Feedback Vertex Set ◽  
Fpt Algorithm ◽  
Vertex Set ◽  
Feedback Vertex Set Problem

On the complexity of alpha conversion

2007 ◽  
Vol 72 (4) ◽  
pp. 1197-1203
Author(s):  
Rick Statman
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Chordal Graphs ◽  
Vertex Coloring ◽  
Feedback Vertex Set ◽  
Vertex Set ◽  
Feedback Vertex Set Problem ◽  
The Way

AbstractWe consider three problems concerning alpha conversion of closed terms (combinators).(1) Given a combinator M find the an alpha convert of M with a smallest number of distinct variables.(2) Given two alpha convertible combinators M and N find a shortest alpha conversion of M to N.(3) Given two alpha convertible combinators M and N find an alpha conversion of M to N which uses the smallest number of variables possible along the way.We obtain the following results.(1) There is a polynomial time algorithm for solving problem (1). It is reducible to vertex coloring of chordal graphs.(2) Problem (2) is co-NP complete (in recognition form). The general feedback vertex set problem for digraphs is reducible to problem (2).(3) At most one variable besides those occurring in both M and N is necessary. This appears to be the folklore but the proof is not familiar. A polynomial time algorithm for the alpha conversion of M to N using at most one extra variable is given.There is a tradeoff between solutions to problem (2) and problem (3) which we do not fully understand.

scholarly journals Recognition and Optimization Algorithms for P5-Free Graphs

Symmetry ◽  
2020 ◽  
Vol 12 (2) ◽  
pp. 304
Author(s):  
Mihai Talmaciu ◽  
Luminiţa Dumitriu ◽  
Ioan Şuşnea ◽  
Victor Lepin ◽  
László Barna Iantovics
Keyword(s):  
Chromatic Number ◽  
Dominating Set ◽  
Independent Set ◽  
Clique Number ◽  
Recognition Algorithm ◽  
Feedback Vertex Set ◽  
Free Graph ◽  
Dominating Number ◽  
Vertex Set ◽  
Np Complete

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.

scholarly journals Clique cycle transversals in graphs with few P₄'s

2013 ◽  
Vol Vol. 15 no. 3 (Graph Theory) ◽  
Author(s):  
Raquel Bravo ◽  
Sulamita Klein ◽  
Loana Tito Nogueira ◽  
Fábio Protti
Keyword(s):  
Linear Time ◽  
Time Algorithm ◽  
Finite Family ◽  
Feedback Vertex Set ◽  
Induced Subgraphs ◽  
Complete Subgraph ◽  
Forbidden Induced Subgraphs ◽  
Vertex Set ◽  
International Audience ◽  
Acyclic Subgraph

Graph Theory International audience A graph is extended P4-laden if each of its induced subgraphs with at most six vertices that contains more than two induced P4's is 2K2,C4-free. A cycle transversal (or feedback vertex set) of a graph G is a subset T ⊆ V (G) such that T ∩ V (C) 6= ∅ for every cycle C of G; if, in addition, T is a clique, then T is a clique cycle transversal (cct). Finding a cct in a graph G is equivalent to partitioning V (G) into subsets C and F such that C induces a complete subgraph and F an acyclic subgraph. This work considers the problem of characterizing extended P4-laden graphs admitting a cct. We characterize such graphs by means of a finite family of forbidden induced subgraphs, and present a linear-time algorithm to recognize them.

scholarly journals Subset feedback vertex set on graphs of bounded independent set size

2020 ◽  
Vol 814 ◽  
pp. 177-188
Author(s):  
Charis Papadopoulos ◽  
Spyridon Tzimas
Keyword(s):  
Independent Set ◽  
Feedback Vertex Set ◽  
Set Size ◽  
Vertex Set

scholarly journals The Parameterised Complexity of Computing the Maximum Modularity of a Graph

Algorithmica ◽  
2019 ◽  
Vol 82 (8) ◽  
pp. 2174-2199 ◽  
Author(s):  
Kitty Meeks ◽  
Fiona Skerman
Keyword(s):  
Vertex Cover ◽  
Constant Factor ◽  
The Other ◽  
Input Graph ◽  
Feedback Vertex Set ◽  
Fpt Algorithm ◽  
Vertex Set ◽  
Fixed Constant ◽  
Minimum Feedback Vertex Set ◽  
Parameterised Complexity

Abstract The maximum modularity of a graph is a parameter widely used to describe the level of clustering or community structure in a network. Determining the maximum modularity of a graph is known to be $$\textsf {NP}$$ NP -complete in general, and in practice a range of heuristics are used to construct partitions of the vertex-set which give lower bounds on the maximum modularity but without any guarantee on how close these bounds are to the true maximum. In this paper we investigate the parameterised complexity of determining the maximum modularity with respect to various standard structural parameterisations of the input graph G. We show that the problem belongs to $$\textsf {FPT}$$ FPT when parameterised by the size of a minimum vertex cover for G, and is solvable in polynomial time whenever the treewidth or max leaf number of G is bounded by some fixed constant; we also obtain an FPT algorithm, parameterised by treewidth, to compute any constant-factor approximation to the maximum modularity. On the other hand we show that the problem is W[1]-hard (and hence unlikely to admit an FPT algorithm) when parameterised simultaneously by pathwidth and the size of a minimum feedback vertex set.

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