Parameterized complexity of the MINCCA problem on graphs of bounded decomposability

In an edge-colored graph, the cost incurred at a vertex on a path when two incident edges with different colors are traversed is called reload or changeover cost. The Minimum Changeover Cost Arborescence (MinCCA) problem consists in finding an arborescence with a given root vertex such that the tota...

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Published in:	Theoretical computer science Vol. 690; pp. 91 - 103
Main Authors:	Gözüpek, Didem, Özkan, Sibel, Paul, Christophe, Sau, Ignasi, Shalom, Mordechai
Format:	Journal Article
Language:	English
Published:	Elsevier B.V 22.08.2017 Elsevier
Subjects:	[formula omitted] algorithm Mathematics Minimum changeover cost arborescence Parameterized complexity Planar graph Tree-cutwidth Treewidth Tree-cutwidth Minimum changeover cost arborescence FPT algorithm Planar graph Parameterized complexity Treewidth
ISSN:	0304-3975, 1879-2294
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Abstract	In an edge-colored graph, the cost incurred at a vertex on a path when two incident edges with different colors are traversed is called reload or changeover cost. The Minimum Changeover Cost Arborescence (MinCCA) problem consists in finding an arborescence with a given root vertex such that the total changeover cost of the internal vertices is minimized. It has been recently proved by Gözüpek et al. (2016) that the MinCCA problem when parameterized by the treewidth and the maximum degree of the input graph is in FPT. In this article we present the following hardness results for MinCCA:•the problem is W[1]-hard when parameterized by the vertex cover number of the input graph, even on graphs of degeneracy at most 3. In particular, it is W[1]-hard parameterized by the treewidth of the input graph, which answers the main open problem in the work of Gözüpek et al. (2016);•it is W[1]-hard on multigraphs parameterized by the tree-cutwidth of the input multigraph; and•it remains NP-hard on planar graphs even when restricted to instances with at most 6 colors and 0/1 symmetric costs, or when restricted to instances with at most 8 colors, maximum degree bounded by 4, and 0/1 symmetric costs.
AbstractList	In an edge-colored graph, the cost incurred at a vertex on a path when two incident edges with different colors are traversed is called reload or changeover cost. The Minimum Changeover Cost Arborescence (MinCCA) problem consists in finding an arborescence with a given root vertex such that the total changeover cost of the internal vertices is minimized. It has been recently proved by Gözüpek et al. (2016) that the MinCCA problem when parameterized by the treewidth and the maximum degree of the input graph is in FPT. In this article we present the following hardness results for MinCCA:•the problem is W[1]-hard when parameterized by the vertex cover number of the input graph, even on graphs of degeneracy at most 3. In particular, it is W[1]-hard parameterized by the treewidth of the input graph, which answers the main open problem in the work of Gözüpek et al. (2016);•it is W[1]-hard on multigraphs parameterized by the tree-cutwidth of the input multigraph; and•it remains NP-hard on planar graphs even when restricted to instances with at most 6 colors and 0/1 symmetric costs, or when restricted to instances with at most 8 colors, maximum degree bounded by 4, and 0/1 symmetric costs.
Author	Gözüpek, Didem Özkan, Sibel Shalom, Mordechai Paul, Christophe Sau, Ignasi
Author_xml	– sequence: 1 givenname: Didem surname: Gözüpek fullname: Gözüpek, Didem email: didem.gozupek@gtu.edu.tr organization: Department of Computer Engineering, Gebze Technical University, Kocaeli, Turkey – sequence: 2 givenname: Sibel orcidid: 0000-0002-9547-7375 surname: Özkan fullname: Özkan, Sibel email: s.ozkan@gtu.edu.tr organization: Department of Mathematics, Gebze Technical University, Kocaeli, Turkey – sequence: 3 givenname: Christophe surname: Paul fullname: Paul, Christophe email: paul@lirmm.fr organization: CNRS, LIRMM, Université de Montpellier, Montpellier, France – sequence: 4 givenname: Ignasi surname: Sau fullname: Sau, Ignasi email: sau@lirmm.fr organization: CNRS, LIRMM, Université de Montpellier, Montpellier, France – sequence: 5 givenname: Mordechai surname: Shalom fullname: Shalom, Mordechai email: cmshalom@telhai.ac.il organization: TelHai College, Upper Galilee, 12210, Israel
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Keywords	Tree-cutwidth Minimum changeover cost arborescence FPT algorithm Planar graph Parameterized complexity Treewidth
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SubjectTerms	[formula omitted] algorithm Mathematics Minimum changeover cost arborescence Parameterized complexity Planar graph Tree-cutwidth Treewidth
Title	Parameterized complexity of the MINCCA problem on graphs of bounded decomposability
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