Finding minimum weighted generators of a path system

In a recent paper the author 1997a] described a short algorithmic proof of a min-max theorem of E. Gy} ori 1984] on minimum generators of a system of subpaths of an underlying path. A. Frank and T. Jordd an 1995] generalized Gy} ori's theorem in several ways; the underlying path was replaced by a circuit, generators were extended to weighted generators, and minimum cost generators were also treated for node-induced cost-functions. The original proof method however was not constructive and the only known algorithm so far to solve the optimization problems in question relied on the ellipsoid method. In this paper a constructive proof is described which gives rise to a purely combinatorial strongly polynomial algorithm.