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

TL;DR

This paper investigates the approximability of multiway partitioning problems, introducing new hardness results and improved approximation algorithms based on symmetry gap and relaxations, especially for submodular and hypergraph cases.

Contribution

It provides the first tight approximation bounds for Submodular Multiway Partition and connects symmetry gap with integrality gap, advancing understanding of hardness and algorithms.

Findings

Lovasz relaxation achieves a (2-2/k)-approximation, improving previous results.

Proves that better than (2-2/k)-approximation requires exponential queries or is NP-hard.

Shows the equivalence of symmetry gap and integrality gap for related problems.

Abstract

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, the Basic LP, and the Lovasz relaxation. We show that the Lovasz relaxation gives a (2-2/k)-approximation for Submodular Multiway Partition with $k$ terminals, improving a recent 2-approximation. We prove that this factor is optimal in two senses: (1) A (2-2/k-\epsilon)-approximation for Submodular Multiway Partition with k terminals would require exponentially many value queries. (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-\epsilon)-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, also implies hardness results for submodular minimization problems. (2) Assuming the Unique Games Conjecture, we show that the Basic 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 Basic 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.