On the approximability of the minimum strictly fundamental cycle basis problem Original Research Article

We consider the problem of finding a strictly fundamental cycle basis of minimum weight in the cycle space associated with an undirected connected graph image, where a nonnegative weight is assigned to each edge of image and the total weight of a basis is defined as the sum of the weights of all the cycles in the basis. Several heuristics have been proposed to tackle this NP-hard problem, which has some interesting applications. In this paper we show that this problem is APX-hard, even when restricted to unweighted graphs, and hence does not admit a polynomial-time approximation scheme, unless image. Using a recent result on the approximability of lower-stretch spanning trees (Elkin et al. (2005) ), we obtain that the problem is approximable within image for arbitrary graphs. We obtain tighter approximability bounds for dense graphs. In particular, the problem restricted to complete graphs admits a polynomial-time approximation scheme.