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When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ISAAC.2016.5
URN: urn:nbn:de:0030-drops-67767
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2016/6776/
Agrawal, Akanksha ;
Panolan, Fahad ;
Saurabh, Saket ;
Zehavi, Meirav
Simultaneous Feedback Edge Set: A Parameterized Perspective
Abstract
In a recent article Agrawal et al. (STACS 2016) studied a simultaneous variant of the classic Feedback Vertex Set problem, called Simultaneous Feedback Vertex Set (Sim-FVS). In this problem the input is an n-vertex graph G, an integer k and a coloring function col : E(G) -> 2^[alpha] , and the objective is to check whether there exists a vertex subset S of cardinality at most k in G such that for all i in [alpha], G_i - S is acyclic. Here, G_i = (V (G), {e in E(G) | i in col(e)}) and [alpha] = {1,...,alpha}. In this paper we consider the edge variant of the problem, namely, Simultaneous Feedback Edge Set (Sim-FES). In this problem, the input is same as the input of Sim-FVS and the objective is to check whether there is an edge subset S of cardinality at most k in G such that for all i in [alpha], G_i - S is acyclic. Unlike the vertex variant of the problem, when alpha = 1, the problem is equivalent to finding a maximal spanning forest and hence it is polynomial time solvable. We show that for alpha = 3 Sim-FES is NP-hard by giving a reduction from Vertex Cover on cubic-graphs. The same reduction shows that the problem does not admit an algorithm of running time O(2^o(k) n^O(1)) unless ETH fails. This hardness result is complimented by an FPT algorithm for Sim-FES running in time O(2^((omega k alpha) + (alpha log k)) n^O(1)), where omega is the exponent in the running time of matrix multiplication. The same algorithm gives a polynomial time algorithm for the case when alpha = 2. We also give a kernel for Sim-FES with (k alpha)^O(alpha) vertices. Finally, we consider the problem Maximum Simultaneous Acyclic Subgraph. Here, the input is a graph G, an integer q and, a coloring function col : E(G) -> 2^[alpha] . The question is whether there is a edge subset F of cardinality at least q in G such that for all i in [alpha], G[F_i] is acyclic. Here, F_i = {e in F | i in col(e)}. We give an FPT algorithm for Maximum Simultaneous Acyclic Subgraph running in time O(2^(omega q alpha) n^O(1) ). All our algorithms are based on parameterized version of the Matroid Parity problem.
BibTeX - Entry
@InProceedings{agrawal_et_al:LIPIcs:2016:6776,
author = {Akanksha Agrawal and Fahad Panolan and Saket Saurabh and Meirav Zehavi},
title = {{Simultaneous Feedback Edge Set: A Parameterized Perspective}},
booktitle = {27th International Symposium on Algorithms and Computation (ISAAC 2016)},
pages = {5:1--5:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-026-2},
ISSN = {1868-8969},
year = {2016},
volume = {64},
editor = {Seok-Hee Hong},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2016/6776},
URN = {urn:nbn:de:0030-drops-67767},
doi = {10.4230/LIPIcs.ISAAC.2016.5},
annote = {Keywords: parameterized complexity, feedback edge set, alpha-matroid parity}
}
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Keywords: |
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parameterized complexity, feedback edge set, alpha-matroid parity |
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Collection: |
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27th International Symposium on Algorithms and Computation (ISAAC 2016) |
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Issue Date: |
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2016 |
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Date of publication: |
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07.12.2016 |
