Separation of partition inequalities

Given a graph G = (V,E) with nonnegative weights x(e) for each edge e, a partition inequality is of the form x(δ(S1,...,Sp))≥ap+b. Here δ(S1,...,Sp) denotes the multicut defined by a partition S1,...,Sp of V. Partition inequalities arise as valid inequalities for optimization problems related to k-connectivity. We give a polynomial algorithm for the associated separation problem. This is based on an algorithm for finding the minimum of x(δ(S1,...,Sp))-p that reduces to minimizing a symmetric submodular function. This is handled with the recent algorithm of Queyranne. We also survey some applications of partition inequalities.