Weighted min cost flows

Abstract

The minimal ratio problem which is, in the context of combinatorial optimization, treated for shortest paths (Dantzig et al. [5]. Fox [10], Karp [12], and Lawler [14,15]) and for spanning trees (Chandrasekaran [4]) is considered in a generalized form for network flow problems. The resulting problem of finding so-called weighted minimal cost flows has nice practical applications: For instance, flows with minimal average cost or flows minimizing cost with respect to possible penalties can be treated in this way. We discuss two algorithms for determining weighted minimal cost flows: the negative circuit algorithm and the shortest augmenting circuit algorithm. The validity of both algorithms follows from a negative circuit theorem for weighted minimal cost flows. A short discussion of the theoretical and computational complexity of the proposed algorithms is closing the paper.