From Graph to Hypergraph Multiway Partition: Is the Single Threshold the Only Route?

We consider the Hypergraph Multiway Partition problem (Hyper-MP). The input consists of an edge-weighted hypergraph G = (V, E) and k vertices s1, . . . , sk called terminals. A multiway partition of the hypergraph is a partition (or labeling) of the vertices of G into k sets A1, . . . , Ak such that si ∈ Ai for each i ∈ [k]. The cost of a multiway partition (A1, . . . , Ak) is ∑k i=1 w(δ(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).