A note on the hardness of approximating the k-way Hypergraph Cut problem

We consider the approximability of k-way Hypergraph Cut problem: the input is an edge-weighted hypergraph G = (V, E) and an integer k and the goal is to remove a min-weight subset of the edges such that the residual graph has at least k connected components. When G is a graph this problem admits a 2(1 − 1/k)-approximation [8], however, there has been no non-trivial approximation ratio for general hypergraphs. In this note we show, via a very simple reduction, that an α-approximation for k-way Hypergraph Cut implies an O(α)approximation for the Densest k-Subgraph problem. This gives conditional hardness of approximation for k-way Hypergraph Cut since the best known approximation ratio for Densest k-Subgraph is O(n) [1] and resolving its approximability is a major open problem. As a corollary we obtain conditional hardness for k-way Submodular Multiway Partition problem which generalizes k-way Hypergraph Cut. These resuls are in contrast to 2(1− 1/k)-approximation for closely related problems where the goal is to separate k given terminals [3, 4].