From graph to Hypergraph Multiway partition: Is the single threshold the only route?

Abstract

We consider the Hypergraph Multiway Partition problem (Hyper-MP). The input consists of an edge-weighted hypergraph and k vertices s 1,⋯, s k called terminals. A multiway partition of the hypergraph is a partition (or labeling) of the vertices of G into k sets A 1, ⋯, A k such that s i A i for each i [k]. The cost of a multiway partition (A 1,⋯, A k ) is ∑ i=jk w( (Ai)), where w(δ(·)) is the hypergraph cut function. The Hyper-MP problem asks for a multiway partition of minimum cost. Our main result is a 4/3 approximation for the Hyper-MP problem on 3-uniform hypergraphs, which is the first improvement over the (1.5-1/k) approximation of [5]. The algorithm combines the single-threshold rounding strategy of Calinescu et al. [3] with the rounding strategy of Kleinberg and Tardos [8], and it parallels the recent algorithm of Buchbinder et al.[2] for the Graph Multiway Cut problem, which is a special case. On the negative side, we show that the KT rounding scheme [8] and the exponential clocks rounding scheme [2] cannot break the (1.5-1/k) barrier for arbitrary hypergraphs. We give a family of instances for which both rounding schemes have an approximation ratio bounded from below by ω (√k), and thus the Graph Multiway Cut rounding schemes may not be sufficient for the Hyper-MP problem when the maximum hyperedge size is large. We remark that these instances have k=Θ(logn). © 2014 Springer-Verlag Berlin Heidelberg.