Local Distribution and the Symmetry Gap: Approximability of Multiway Partitioning Problems

We study the approximability of multiway partitioning problems, examples of which include Multiway Cut, Node-weighted Multiway Cut, and Hypergraph Multiway Cut. We investigate these problems from the point of view of two possible generalizations: as Min-CSPs, and as Submodular Multiway Partition problems. These two generalizations lead to two natural relaxations that we call respectively the Local Distribution LP, and the Lovász relaxation. The Local Distribution LP is generally stronger than the Lovász relaxation, but applicable only to Min-CSP with predicates of constant size. The relaxations coincide in some cases such as Multiway Cut where they are both equivalent to the CKR relaxation. We show that the Lovász relaxation gives a (2 − 2/k)-approximation for Submodular Multiway Partition with k terminals, improving a recent 2-approximation [2]. We prove that this factor is optimal in two senses: (1) A (2−2/k− )-approximation for Submodular Multiway Partition with k terminals would require exponentially many value queries (in the oracle model), or imply NP = RP (for certain explicit subomdular functions). (2) For Hypergraph Multiway Cut and Node-weighted Multiway Cut with k terminals, both special cases of Submodular Multiway Partition, we prove that a (2−2/k− )-approximation is NP-hard, assuming the Unique Games Conjecture. Both our hardness results are more general: (1) We show that the notion of symmetry gap, previously used for submodular maximization problems [19, 6], implies hardness results also for submodular minimization problems. (2) Assuming the Unique Games Conjecture, we show that the Local Distribution LP gives an optimal approximation for every Min-CSP that includes the Not-Equal predicate. Finally, we connect the two hardness techniques by proving that the integrality gap of the Local Distribution LP coincides with the symmetry gap of the multilinear relaxation (for a related instance). This shows that the appearance of the same hardness threshold for a Min-CSP and the related submodular minimization problem is not a coincidence. ∗Dept. of Computer Science, University of Illinois, Urbana-Champaign, IL †IBM Almaden Research Center, San Jose, CA ‡IBM Almaden Research Center, San Jose, CA