On fractional cut covers

Given an undirected graph, a minimum cut cover is a collection of cuts covering the whole set of edges and having minimum cardinality. This paper is dedicated to the fractional version of this problem where a fractional weight is computed for each cut such that, for each edge, the sum of the weights of all cuts containing it is no less than 1, while the sum of all weights is minimized. The fractional cover is computed for different graph classes among which are the weakly bipartite graphs. Efficient algorithms are described to compute lower and upper bounds with worst-case performance guarantees. A general randomized approach is also presented giving new insights into Goemans and Williamson's algorithm for the maximum cut problem. Some numerical experiments are included to assess the quality of bounds.