Covering symmetric supermodular functions by graphs

The minimum number of edges of an undirected graph covering a symmetric , supermodular set-function is determined. As a special case, we derive an extension of a theorem of J. Bang-Jensen and B. Jackson on hypergraph connectivity augmentation. 0. INTRODUCTION T. Watanabe and A. Nakamura 1987] proved a min-max formula for the minimum number of new edges whose addition to a given undirected graph results in a k-edge-connected graph. E. Cheng ] considered the problem of increasing the connectivity of hypergraphs by adding a minimum number of graph-edges, and provided a solution for the special case when the starting hypergraph is (k ? 1)-edge-connected. Extending further the results of Cheng, J. Bang-Jensen and B. Jackson ] solved the general hypergraph connectivity augmentation problem. The purpose of the present paper is to derive a generalization of the theorem of Bang-Jensen and Jackson where, instead of a hypergraph whose connectivity is to be incereased, a symmetric supermodular function is speciied to bècovered' by undirected edges. The result, when specialized to hypergraphs, not only provides the theorem of Bang-Jensen and Jackson, but it actually gives rise to an extension when the connectivity of the hypergraph is to be increased inside a speciied terminal set. This will be explained in Section 6. Let us say some words about the proof methods. For proving their theorem, Watanabe and Nakamura used a sophisticated analysis of the structure of k-edge-connected graphs. A diierent proof, based on the splitting-oo technique, was given by G-R. Cai and Y-G. Sun 1989]. A. Frank 1992] used the the splitting-oo technique in a diierent way and obtained a short proof of the theorem of Watanabe and Nakamura. This simpliication enabled him to nd several extensions such as local-edge-connectivity augmentation, minimum node-cost and degree-constrained augmentation. The same technique gave rise to a directed counterpart of the theorem of Watanabe and Nakamura. The hypergraph connectivity augmentation problem is signiicantly more diicult than the graph augmentation problem since a new type of necessary conditions comes in. This was recognized by E. Cheng whose method for increasing optimally the edge-connectivity of a hypergraph by one is based on a decomposition technique of submodular functions due to W. Cunningham 1983]. For solving the general hypergraph augmentation problem, Bang-Jensen and Jackson returned to the splitting technique but they also needed the inverse operation of splitting. The present method stems out of the work of Bang-Jensen and Jackson. Our …