We obtain a necessary and sufficient condition in terms of forbidden structures for tournaments to possess the min-max relation on packing and covering directed cycles, together with strongly polynomial time algorithms for the feedback vertex set problem and the cycle packing problem in this class of tournaments. Applying the local ratio technique of Bar-Yehuda and Even to the forbidden structures, we find a 2.5-approximation polynomial time algorithm for the feedback vertex set problem in any tournament.

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Let A be a 0 − 1 matrix with precisely two 1's in each column and let 1 be the all-one vector. We show that the problems of deciding whether the linear system Ax ≥ 1, x ≥ 0 (1) defines an integral polyhedron, (2) is totally dual integral (TDI), and (3) is box-totally dual integral (box-TDI) are all co-NP-complete, thereby confirming the conjecture on NP-hardness of recognizing TDI systems made by Edmonds and Giles in 1984.

A graph G is called cycle Mengerian (CM) if for all nonnegative integral function w defined on V ðGÞ; the maximum number of cycles (repetition is allowed) in G such that each vertex v is used at most wðvÞ times is equal to the minimum of P fwðxÞ : x 2 X g; where the minimum is taken over all X V ðGÞ such that deleting X from G results in a forest. The purpose of this paper is to characterize all CM graphs in terms of forbidden structures. As a corollary, we prove that if the fractional version of the above minimization problem always have an integral optimal solution, then the fractional version of the maximization problem will always have an integral optimal solution as well. # 2002 Elsevier Science (USA)