Finding cheapest cycles in vertex-weighted quasi-transitive and extended semicomplete digraphs

Authors:
JøRgen Bang-Jensen;Gregory Gutin;Anders Yeo
Affiliations:
Department of Mathematics and Computer Science, University of Southern Denmark, Odense M, DK-5230, Denmark;Department of Computer Science, Royal Holloway, University of London, Egham, Surrey TW20 0EX, UK;Department of Computer Science, Royal Holloway, University of London, Egham, Surrey TW20 0EX, UK
Venue:
Discrete Optimization
Year:
2006

Citing 5
Cited 1

Quasi-transitive digraphs

Journal of Graph Theory
Vertex heaviest paths and cycles in quasi-transitive digraphs

Discrete Mathematics
A polynomial time algorithm for finding a cycle covering a given set of vertices in a semicomplete multipartite digraph

Journal of Algorithms
Orientations of digraphs almost preserving diameter

Discrete Applied Mathematics
Steiner type problems for digraphs that are locally semicomplete or extended semicomplete

Journal of Graph Theory

Minimum cycle factors in quasi-transitive digraphs

Discrete Optimization

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Abstract

We consider the problem of finding a minimum cost cycle in a digraph with real-valued costs on the vertices. This problem generalizes the problem of finding a longest cycle and hence is NP-hard for general digraphs. We prove that the problem is solvable in polynomial time for extended semicomplete digraphs and for quasi-transitive digraphs, thereby generalizing a number of previous results on these classes. As a byproduct of our method we develop polynomial algorithms for the following problem: Given a quasi-transitive digraph D with real-valued vertex costs, find, for each j=1,2,...,|V(D)|, j disjoint paths P"1,P"2,...,P"j such that the total cost of these paths is minimum among all collections of j disjoint paths in D.