Approximation Algorithms for Requirement Cut on Graphs

In this paper, we unify several graph partitioning problems including multicut, multiway cut, and k-cut, into a single problem. The input to the requirement cut problem is an undirected edge-weighted graph G=(V,E), and g groups of vertices X (1),aEuro broken vertical bar,X (g) aS dagger V, with each group X (i) having a requirement r (i) between 0 and |X (i) |. The goal is to find a minimum cost set of edges whose removal separates each group X (i) into at least r (i) disconnected components. We give an O(log na <...log (gR)) approximation algorithm for the requirement cut problem, where n is the total number of vertices, g is the number of groups, and R is the maximum requirement. We also show that the integrality gap of a natural LP relaxation for this problem is bounded by O(log na <...log (gR)). On trees, we obtain an improved guarantee of O(log (gR)). There is an Omega(log g) hardness of approximation for the requirement cut problem, even on trees.