THE MINIMUM SPANNING SUBGRAPH PROBLEM WITH GIVEN CYCLOMATIC NUMBER

This paper discusses the minimum spanning subgraph problem with given cyclomatic number k, where the cyclomatic number of a graph G, denoted by beta(G), is the dimension of its cycle space. For a given weighted graph G = (V, E, w), the problem asks to find a spanning subgraph F of G such that beta(F) = k and w(F) is as small as possible. We show that this problem is strongly NP-hard. When both G and F are required to be connected, we present a strongly polynomial-time algorithm to solve this problem. In this case, we also consider its reverse problem, which asks how to modify the weight function w under some given bounds in graph G such that the total modification cost plus the total weight of F under the new weights is minimized. A strongly polynomial-time algorithm is also proposed to solve this reverse problem.