A set function on the power set of a finite set is said to be symmetric if the value for each subset coincides with that for its complement. A cut capacity function of an undirected graph or hypergraph is a fundamental example of a symmetric set function, and is also submodular if the capacity on each edge or hyperedge is nonnegative. Fujishige and Patkar (2001) provided necessary and sufficient conditions for set functions to be realized as cut capacity functions of several types of networks. In this paper, we focus on the case of undirected hypergraphs, which was not dealt with in the previous work of Fujishige and Patkar. For this case, Grishuhin (1989) had given a set of hyperedges forming a basis of the cut realization of symmetric set functions. By using the Möbius inversion formula together with Grishuhin's basis, we extend a result of Fujishige and Patkar for the case of undirected graphs to hypergraphs. We also clarify the kernel of the cut realization, which leads to other interesting bases and an alternative proof of Grishuhin's result. In addition, we provide a new necessary condition for the cut realization by undirected hypergraphs with nonnegative capacity.
All Science Journal Classification (ASJC) codes
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics