Network synthesis problem is the problem of constructing a minimum cost network satisfying a given flow-requirement. A classical result of Gomory and Hu is that if the cost is uniform and the flow requirement is integer-valued, then there exists a half-integral optimal solution. They also gave a simple algorithm to find a half-integral optimal solution. In this paper, we show that this half-integrality and the Gomory-Hu algorithm can be extended to a class of fractional cut-covering problems defined by skew-supermodular functions. Application to approximation algorithm is also given.

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For the problem of enumerating cuts of graphs, Provan and Shier have provided a framework of enumeration that can be applied for the enumeration of various types of cuts such as minimal (s, t)-cuts in an undirected graph, minimal (s, t)-cuts in a directed graph, minimal (s, K)-cuts in an undirected graph, etc. In this paper we generalize their framework and provide an enumerating method that works in a looser condition, providing a possibility of its application easier. We demonstrate its use by giving an algorithm for the enumeration problem that contains the problem of listing minimal (s, K)-cuts in an undirected graph.

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