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

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 DD with real-valued vertex costs, find, for each j=1,2,…,|V(D)|j=1,2,…,|V(D)|, jj disjoint paths P1,P2,…,PjP1,P2,…,Pj such that the total cost of these paths is minimum among all collections of jj disjoint paths in DD.