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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Bibliographic Details
Published in:Discrete Applied Mathematics Vol. 159; no. 16; pp. 1726 - 1735
Main Authors: Darmann, Andreas, Pferschy, Ulrich, Schauer, Joachim, Woeginger, Gerhard J.
Format: Journal Article Conference Proceeding
Language:English
Published: Kidlington Elsevier B.V 28.09.2011
Elsevier
Subjects:
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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Summary: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.
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ISSN:0166-218X
1872-6771
DOI:10.1016/j.dam.2010.12.016