In the constrained max k-cut problem on hypergraphs, we are given a weighted hypergraph H=(V, E), an integer k and a set c of constraints. The goal is to divide the set V of vertices into k disjoint partitions in such a way that the sum of the weights of the hyperedges having at least two endpoints in different partitions is maximized and the partitions satisfy all the constraints in c. In this paper we present a local search algorithm for the constrained max k-cut problem on hypergraphs and show that it has approximation ratio 1-1/k for a variety of constraints c, such as for the constraints defining the max Steiner k-cut problem, the max multiway cut problem and the max k-cut problem. We also show that our local search algorithm can be used on the max k-cut problem with given sizes of parts and on the capacitated max k-cut problem, and has approximation ratio 1-|Vmax|/|V|, where |Vmax| is the cardinality of the biggest partition. In addition, we present a local search algorithm for the directed max k-cut problem that has approximation ratio (k-1)/(3k-2).
A local search approximation algorithm for k-means clustering
