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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Vydáno v:Algorithmica Ročník 76; číslo 1; s. 279 - 296
Hlavní autoři: Bang-Jensen, Jørgen, Saurabh, Saket, Simonsen, Sven
Médium: Journal Article
Jazyk:angličtina
Vydáno: New York Springer US 01.09.2016
Témata:
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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Shrnutí: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.
ISSN:0178-4617
1432-0541
DOI:10.1007/s00453-015-0037-3