scholarly journals 2-Approximating Feedback Vertex Set in Tournaments

2021 ◽  
Vol 17 (2) ◽  
pp. 1-14
Author(s):  
Daniel Lokshtanov ◽  
Pranabendu Misra ◽  
Joydeep Mukherjee ◽  
Fahad Panolan ◽  
Geevarghese Philip ◽  
...  
Keyword(s):  
Approximation Algorithm ◽  
Weight Function ◽  
Directed Graph ◽  
Polynomial Time ◽  
Optimal Solutions ◽  
Feedback Vertex Set ◽  
Vertex Set ◽  
Feedback Vertex Set Problem ◽  
Unique Games Conjecture ◽  
Unique Games

A tournament is a directed graph T such that every pair of vertices is connected by an arc. A feedback vertex set is a set S of vertices in T such that T − S is acyclic. We consider the Feedback Vertex Set problem in tournaments. Here, the input is a tournament T and a weight function w : V ( T ) → N, and the task is to find a feedback vertex set S in T minimizing w ( S ) = ∑ v∈S w ( v ). Rounding optimal solutions to the natural LP-relaxation of this problem yields a simple 3-approximation algorithm. This has been improved to 2.5 by Cai et al. [SICOMP 2000], and subsequently to 7/3 by Mnich et al. [ESA 2016]. In this article, we give the first polynomial time factor 2-approximation algorithm for this problem. Assuming the Unique Games Conjecture, this is the best possible approximation ratio achievable in polynomial time.

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.

An 8-Approximation Algorithm for the Subset Feedback Vertex Set Problem

2000 ◽  
Vol 30 (4) ◽  
pp. 1231-1252 ◽  
Author(s):  
Guy Even ◽  
Joseph (Seffi) Naor ◽  
Leonid Zosin
Keyword(s):  
Approximation Algorithm ◽  
Feedback Vertex Set ◽  
Vertex Set ◽  
Feedback Vertex Set Problem

Parameterized complexity of fair feedback vertex set problem

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

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.

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