Finding a cheapest
cycle in a quasi-transitive digraph with real-valued vertex costs ## AbstractWe 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 $j=1,2,\ldots{},|V(D)|$, $j$ disjoint paths $P_1,P_2,\ldots{},P_j$ such that the total cost of these paths is minimum among all collections of $j$ disjoint paths in $D$.
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