On min-max cycle bases

An undirected biconnected graph G with non negative weights on the edges is given. In the cycle space associated with G, a subspace of the vector space of G, we define as weight of a basis the maximum among the weights of the cycles of the basis. The problem we consider is that of finding a basis of minimum weight for the cycle space. It is easy to see that if we do not put additional constraints on the basis, then the problem is easy and there are fast algorithms for solving it. On the other hand if we require the basis to be fundamental, i.e. to consist of the set of all fundamental cycles of G with respect to the chords of a spanning tree of G, then we show that the problem is NP-hard and cannot be approximated within 2 - is an element of,For Allis an element of > 0, even with uniform weights, unless P=NP. We also show that the problem remains NP-hard when restricted to the class of complete graphs; in this case it cannot be approximated within 13/11 - is an element of,For Allis an element of > 0, unless P=NP; it is instead approximable within 2 in general, and within 3/2 if the triangle inequality holds.