Paths, trees and matchings under disjunctive constraints

We study the minimum spanning tree problem, the maximum matching problem and the shortest path problem subject to binary disjunctive constraints: A negative disjunctive constraint states that a certain pair of edges cannot be contained simultaneously in a feasible solution. It is convenient to repre...

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Veröffentlicht in:	Discrete Applied Mathematics Jg. 159; H. 16; S. 1726 - 1735
Hauptverfasser:	Darmann, Andreas, Pferschy, Ulrich, Schauer, Joachim, Woeginger, Gerhard J.
Format:	Journal Article Tagungsbericht
Sprache:	Englisch
Veröffentlicht:	Kidlington Elsevier B.V 28.09.2011 Elsevier
Schlagworte:	Algorithmics. Computability. Computer arithmetics Applied sciences Binary constraints Combinatorics Combinatorics. Ordered structures Complexity Computer science; control theory; systems Conflict graph Exact sciences and technology Graph theory Graphs Information retrieval. Graph Matching Mathematical analysis Mathematics Minimal spanning tree Sciences and techniques of general use Shortest path Shortest-path problems Theoretical computing Trees Matching Conflict graph Shortest path Minimal spanning tree Binary constraints Edge(graph) Graph path Computer theory Connected graph Optimization Complexity Maximum Spanning tree Combinatorics State constraint
ISSN:	0166-218X, 1872-6771
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Abstract	We study the minimum spanning tree problem, the maximum matching problem and the shortest path problem subject to binary disjunctive constraints: A negative disjunctive constraint states that a certain pair of edges cannot be contained simultaneously in a feasible solution. It is convenient to represent these negative disjunctive constraints in terms of a so-called conflict graph whose vertices correspond to the edges of the underlying graph, and whose edges encode the constraints. We prove that the minimum spanning tree problem is strongly NP -hard, even if every connected component of the conflict graph is a path of length two. On the positive side, this problem is polynomially solvable if every connected component is a single edge (that is, a path of length one). The maximum matching problem is NP -hard for conflict graphs where every connected component is a single edge. Furthermore we will also investigate these graph problems under positive disjunctive constraints: In this setting for certain pairs of edges, a feasible solution must contain at least one edge from every pair. We establish a number of complexity results for these variants including APX-hardness for the shortest path problem.
AbstractList	We study the minimum spanning tree problem, the maximum matching problem and the shortest path problem subject to binary disjunctive constraints: A negative disjunctive constraint states that a certain pair of edges cannot be contained simultaneously in a feasible solution. It is convenient to represent these negative disjunctive constraints in terms of a so-called conflict graph whose vertices correspond to the edges of the underlying graph, and whose edges encode the constraints. We prove that the minimum spanning tree problem is strongly [inline image][inline image]-hard, even if every connected component of the conflict graph is a path of length two. On the positive side, this problem is polynomially solvable if every connected component is a single edge (that is, a path of length one). The maximum matching problem is [inline image][inline image]-hard for conflict graphs where every connected component is a single edge. Furthermore we will also investigate these graph problems under positive disjunctive constraints: In this setting for certain pairs of edges, a feasible solution must contain at least one edge from every pair. We establish a number of complexity results for these variants including APX-hardness for the shortest path problem. We study the minimum spanning tree problem, the maximum matching problem and the shortest path problem subject to binary disjunctive constraints: A negative disjunctive constraint states that a certain pair of edges cannot be contained simultaneously in a feasible solution. It is convenient to represent these negative disjunctive constraints in terms of a so-called conflict graph whose vertices correspond to the edges of the underlying graph, and whose edges encode the constraints. We prove that the minimum spanning tree problem is strongly NP -hard, even if every connected component of the conflict graph is a path of length two. On the positive side, this problem is polynomially solvable if every connected component is a single edge (that is, a path of length one). The maximum matching problem is NP -hard for conflict graphs where every connected component is a single edge. Furthermore we will also investigate these graph problems under positive disjunctive constraints: In this setting for certain pairs of edges, a feasible solution must contain at least one edge from every pair. We establish a number of complexity results for these variants including APX-hardness for the shortest path problem.
Author	Schauer, Joachim Darmann, Andreas Woeginger, Gerhard J. Pferschy, Ulrich
Author_xml	– sequence: 1 givenname: Andreas surname: Darmann fullname: Darmann, Andreas email: andreas.darmann@uni-graz.at organization: University of Graz, Institute of Public Economics, Universitaetsstr. 15, A-8010 Graz, Austria – sequence: 2 givenname: Ulrich surname: Pferschy fullname: Pferschy, Ulrich email: pferschy@uni-graz.at organization: University of Graz, Department of Statistics and Operations Research, Universitaetsstr. 15, A-8010 Graz, Austria – sequence: 3 givenname: Joachim surname: Schauer fullname: Schauer, Joachim email: joachim.schauer@uni-graz.at organization: University of Graz, Department of Statistics and Operations Research, Universitaetsstr. 15, A-8010 Graz, Austria – sequence: 4 givenname: Gerhard J. surname: Woeginger fullname: Woeginger, Gerhard J. email: gwoegi@win.tue.nl organization: TU Eindhoven, Department of Mathematics and Computer Science, P.O. Box 513, NL-5600 MB Eindhoven, The Netherlands
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Cites_doi	10.7155/jgaa.00186 10.1016/S0167-5060(08)70817-3 10.1006/inco.1996.2616 10.1023/A:1009871302966
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Keywords	Matching Conflict graph Shortest path Minimal spanning tree Binary constraints Edge(graph) Graph path Computer theory Connected graph Optimization Complexity Maximum Spanning tree Combinatorics State constraint
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SubjectTerms	Algorithmics. Computability. Computer arithmetics Applied sciences Binary constraints Combinatorics Combinatorics. Ordered structures Complexity Computer science; control theory; systems Conflict graph Exact sciences and technology Graph theory Graphs Information retrieval. Graph Matching Mathematical analysis Mathematics Minimal spanning tree Sciences and techniques of general use Shortest path Shortest-path problems Theoretical computing Trees
Title	Paths, trees and matchings under disjunctive constraints
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