Partition Constrained Covering of a Symmetric Crossing Supermodular Function by a Graph

We are given a symmetric crossing supermodular set function $p$ on $V$ and a partition $\mathcal{P}$ of $V$. We solve the problem of finding a graph with vertex set $V$ having edges only between the classes of $\mathcal{P}$ such that for every subset $X$ of $V$ the cut of the graph defined by $X$ contains at least $p(X)$ edges. The objective is to minimize the number of edges of the graph. This problem is a common generalization of the global edge-connectivity augmentation of a graph with partition constraints, which was solved by Bang-Jensen et al. [SIAM J. Discrete Math., 12 (1999), pp. 160--207] and the problem of covering a symmetric crossing supermodular set function solved by Benczur and Frank [Math. Program., 84 (1999), pp. 483--503]. Our problem can be considered as an abstract form of the problem of global edge-connectivity augmentation of a hypergraph with partition constraints, which was earlier solved by the authors [J. Graph Theory, 72 (2013), pp. 291--312].