Paths, trees and matchings under disjunctive constraints

• Published in: Discrete Applied Mathematics 2011-09, Vol.159 (16), p.1726-1735
• Main Authors: Darmann, Andreas, Pferschy, Ulrich, Schauer, Joachim, Woeginger, Gerhard J.
• Format: Article
• Language: English
• 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
• ISSN: 0166-218X; 1872-6771
• DOI: 10.1016/j.dam.2010.12.016

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.