Parameterized Algorithms for Non-separating Trees and Branchings in Digraphs

A well known result in graph algorithms, due to Edmonds, states that given a digraph D and a positive integer ℓ , we can test whether D contains ℓ arc-disjoint out-branchings in polynomial time. However, if we ask whether there exists an out-branching and an in-branching which are arc-disjoint, then...

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Published in:	Algorithmica Vol. 76; no. 1; pp. 279 - 296
Main Authors:	Bang-Jensen, Jørgen, Saurabh, Saket, Simonsen, Sven
Format:	Journal Article
Language:	English
Published:	New York Springer US 01.09.2016
Subjects:	Algorithm Analysis and Problem Complexity Algorithms Computer Science Computer Systems Organization and Communication Networks Data Structures and Information Theory Mathematics of Computing Theory of Computation Branching Linear vertex kernel Partitioning problem Spanning tree Fixed parameter tractable Parameterized complexity Exponential time algorithm
ISSN:	0178-4617, 1432-0541
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Abstract	A well known result in graph algorithms, due to Edmonds, states that given a digraph D and a positive integer ℓ , we can test whether D contains ℓ arc-disjoint out-branchings in polynomial time. However, if we ask whether there exists an out-branching and an in-branching which are arc-disjoint, then the problem becomes NP-complete. In fact, even deciding whether a digraph D contains an out-branching which is arc-disjoint from some spanning tree in the underlying undirected graph remains NP-complete. In this paper we formulate some natural optimization questions around these problems and initiate its study in the realm of parameterized complexity. More precisely, the problems we study are the following: Arc - Disjoint Branchings and Non - Disconnecting Out - Branching . In Arc - Disjoint Branchings ( Non - Disconnecting Out - Branching ), a digraph D and a positive integer k are given as input and the goal is to test whether there exist an out-branching and in-branching (respectively, a spanning tree in the underlying undirected graph) that differ on at least k arcs. We obtain the following results for these problems. Non - Disconnecting Out - Branching is fixed parameter tractable (FPT) and admits a linear vertex kernel. Arc - Disjoint Branchings is FPT on strong digraphs. The algorithm for Non - Disconnecting Out - Branching runs in time 2 O ( k ) n O ( 1 ) and the approach we use to obtain this algorithms seems useful in designing other moderately exponential time algorithms for edge/arc partitioning problems.
AbstractList	A well known result in graph algorithms, due to Edmonds, states that given a digraph D and a positive integer ℓ , we can test whether D contains ℓ arc-disjoint out-branchings in polynomial time. However, if we ask whether there exists an out-branching and an in-branching which are arc-disjoint, then the problem becomes NP-complete. In fact, even deciding whether a digraph D contains an out-branching which is arc-disjoint from some spanning tree in the underlying undirected graph remains NP-complete. In this paper we formulate some natural optimization questions around these problems and initiate its study in the realm of parameterized complexity. More precisely, the problems we study are the following: Arc - Disjoint Branchings and Non - Disconnecting Out - Branching . In Arc - Disjoint Branchings ( Non - Disconnecting Out - Branching ), a digraph D and a positive integer k are given as input and the goal is to test whether there exist an out-branching and in-branching (respectively, a spanning tree in the underlying undirected graph) that differ on at least k arcs. We obtain the following results for these problems. Non - Disconnecting Out - Branching is fixed parameter tractable (FPT) and admits a linear vertex kernel. Arc - Disjoint Branchings is FPT on strong digraphs. The algorithm for Non - Disconnecting Out - Branching runs in time 2 O ( k ) n O ( 1 ) and the approach we use to obtain this algorithms seems useful in designing other moderately exponential time algorithms for edge/arc partitioning problems.
Author	Simonsen, Sven Bang-Jensen, Jørgen Saurabh, Saket
Author_xml	– sequence: 1 givenname: Jørgen surname: Bang-Jensen fullname: Bang-Jensen, Jørgen organization: Department of Mathematics and Computer Science, University of Southern Denmark – sequence: 2 givenname: Saket surname: Saurabh fullname: Saurabh, Saket email: saket@imsc.res.in organization: The Institute of Mathematical Sciences – sequence: 3 givenname: Sven surname: Simonsen fullname: Simonsen, Sven organization: Department of Mathematics and Computer Science, University of Southern Denmark
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Keywords	Branching Linear vertex kernel Partitioning problem Spanning tree Fixed parameter tractable Parameterized complexity Exponential time algorithm
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SubjectTerms	Algorithm Analysis and Problem Complexity Algorithms Computer Science Computer Systems Organization and Communication Networks Data Structures and Information Theory Mathematics of Computing Theory of Computation
Title	Parameterized Algorithms for Non-separating Trees and Branchings in Digraphs
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