New heuristics for the minimum fundamental cut basis problem

Given an undirected connected network G = (V,E), n = |V |, m = |E|, and a weight function w : E 7→ R+, finding a basis of the cut space with minimum sum of the cut weights is termed Minimum Cut Basis Problem. This problem can be solved, e.g., by the algorithm of Gomory and Hu [GH61]. If, however, fundamentality is required, i.e., the basis is induced by a spanning tree T in G, the problem becomes NP-hard. Theoretical and numerical results on that topic can be found in Bunke et al. [BHMM07] and in Bunke [Bun06]. In the following we present heuristics with complexity O(m log n) and O(mn) which obtain upper bounds on the aforementioned problem and in several cases outperform the heuristics of Schwahn [Sch05].