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 $\mathcal{G} = (V, \epsilon)$ and k vertices s ₁, …, s _k called terminals. A multiway partition of the hypergraph is a partition (or labeling) of the vertices of $\mathcal{G}$ 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 $\sum_{i = 1}^k w(\delta(A_i))$, where $w(\delta(\cdotp))$ 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 $\Omega(\sqrt{k})$, 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).