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. Gyori [1984] on minimum generators of a system of subpaths of an underlying path. A. Frank and T. Jord'an [1995] generalized Gyori'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. 1. INTRODUCTION Let P = (v 0 ; j 1 ; v 1 ; j 2 ; v 2 ; : : : ; j n ; vn = v 0 ) be a directed circuit, that is, each directed edge j i has tail v i\Gamma1 and head v i , and the nodes v 1 ; : : : ; v n are distinct. Let V := fv 0 ; : : : ; v ng and E := fj 1 ; : : : ; j ng ..