scholarly journals FPT Algorithms for Diverse Collections of Hitting Sets

Algorithms ◽  
2019 ◽  
Vol 12 (12) ◽  
pp. 254
Author(s):  
Julien Baste ◽  
Lars Jaffke ◽  
Tomáš Masařík ◽  
Geevarghese Philip ◽  
Günter Rote
Keyword(s):  
Network Flow ◽  
Hamming Distance ◽  
Side Information ◽  
Feedback Vertex Set ◽  
Fixed Parameter Tractable ◽  
Fpt Algorithms ◽  
Fixed Parameter ◽  
Vertex Set ◽  
Two Measures ◽  
Real World Problems

In this work, we study the d-Hitting Set and Feedback Vertex Set problems through the paradigm of finding diverse collections of r solutions of size at most k each, which has recently been introduced to the field of parameterized complexity. This paradigm is aimed at addressing the loss of important side information which typically occurs during the abstraction process that models real-world problems as computational problems. We use two measures for the diversity of such a collection: the sum of all pairwise Hamming distances, and the minimum pairwise Hamming distance. We show that both problems are fixed-parameter tractable in k + r for both diversity measures. A key ingredient in our algorithms is a (problem independent) network flow formulation that, given a set of ‘base’ solutions, computes a maximally diverse collection of solutions. We believe that this could be of independent interest.

scholarly journals Subset Feedback Vertex Set Is Fixed-Parameter Tractable

2013 ◽  
Vol 27 (1) ◽  
pp. 290-309 ◽  
Author(s):  
Marek Cygan ◽  
Marcin Pilipczuk ◽  
Michał Pilipczuk ◽  
Jakub Onufry Wojtaszczyk
Keyword(s):  
Feedback Vertex Set ◽  
Fixed Parameter Tractable ◽  
Fixed Parameter ◽  
Vertex Set

scholarly journals Subset Feedback Vertex Set Is Fixed-Parameter Tractable

2011 ◽  
pp. 449-461 ◽  
Author(s):  
Marek Cygan ◽  
Marcin Pilipczuk ◽  
Michał Pilipczuk ◽  
Jakub Onufry Wojtaszczyk
Keyword(s):  
Feedback Vertex Set ◽  
Fixed Parameter Tractable ◽  
Fixed Parameter ◽  
Vertex Set

scholarly journals Directed Subset Feedback Vertex Set Is Fixed-Parameter Tractable

2015 ◽  
Vol 11 (4) ◽  
pp. 1-28 ◽  
Author(s):  
Rajesh Chitnis ◽  
Marek Cygan ◽  
Mohammataghi Hajiaghayi ◽  
Dániel Marx
Keyword(s):  
Feedback Vertex Set ◽  
Fixed Parameter Tractable ◽  
Fixed Parameter ◽  
Vertex Set

scholarly journals Towards a Polynomial Kernel for Directed Feedback Vertex Set

Algorithmica ◽  
2020 ◽  
Author(s):  
Benjamin Bergougnoux ◽  
Eduard Eiben ◽  
Robert Ganian ◽  
Sebastian Ordyniak ◽  
M. S. Ramanujan
Keyword(s):  
Undirected Graph ◽  
Polynomial Kernel ◽  
Directed Cycle ◽  
Feedback Vertex Set ◽  
Open Problems ◽  
Fixed Parameter Tractable ◽  
Parameterized Algorithmics ◽  
Fixed Parameter ◽  
Vertex Set ◽  
Directed Feedback Vertex Set

Abstract In the Directed Feedback Vertex Set (DFVS) problem, the input is a directed graph D and an integer k. The objective is to determine whether there exists a set of at most k vertices intersecting every directed cycle of D. DFVS was shown to be fixed-parameter tractable when parameterized by solution size by Chen et al. (J ACM 55(5):177–186, 2008); since then, the existence of a polynomial kernel for this problem has become one of the largest open problems in the area of parameterized algorithmics. Since this problem has remained open in spite of the best efforts of a number of prominent researchers and pioneers in the field, a natural step forward is to study the kernelization complexity of DFVS parameterized by a natural larger parameter. In this paper, we study DFVS parameterized by the feedback vertex set number of the underlying undirected graph. We provide two main contributions: a polynomial kernel for this problem on general instances, and a linear kernel for the case where the input digraph is embeddable on a surface of bounded genus.

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.

scholarly journals FPT Algorithms for Connected Feedback Vertex Set

2010 ◽  
pp. 269-280 ◽  
Author(s):  
Neeldhara Misra ◽  
Geevarghese Philip ◽  
Venkatesh Raman ◽  
Saket Saurabh ◽  
Somnath Sikdar
Keyword(s):  
Feedback Vertex Set ◽  
Fpt Algorithms ◽  
Vertex Set

scholarly journals FPT-ALGORITHMS FOR MINIMUM-BENDS TOURS

2011 ◽  
Vol 21 (02) ◽  
pp. 189-213 ◽  
Author(s):  
VLADIMIR ESTIVILL-CASTRO ◽  
APICHAT HEEDNACRAM ◽  
FRANCIS SURAWEERA
Keyword(s):  
Traveling Salesman Problem ◽  
Traveling Salesman ◽  
Fixed Parameter Tractable ◽  
Line Segments ◽  
Fpt Algorithms ◽  
Fixed Parameter ◽  
Axis Parallel ◽  
Np Complete ◽  
Number Of Bends ◽  
Straight Lines

This paper discusses the κ-BENDS TRAVELING SALESMAN PROBLEM. In this NP-complete problem, the inputs are n points in the plane and a positive integer κ, and we are asked whether we can travel in straight lines through these n points with at most κ bends. There are a number of applications where minimizing the number of bends in the tour is desirable because bends are considered very costly. We prove that this problem is fixed-parameter tractable (FPT). The proof is based on the kernelization approach. We also consider the RECTILINEAR κ-BENDS TRAVELING SALESMAN PROBLEM, which requires that the line-segments be axis-parallel. 1 Note that a rectilinear tour with κ bends is a cover with κ-line segments, and therefore a cover by lines. We introduce two types of constraints derived from the distinction between line-segments and lines. We derive FPT-algorithms with different techniques and improved time complexity for these cases.

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