On the Computational Complexity of Finding a Sparse Wasserstein Barycenter

TL;DR

This paper demonstrates that computing sparse Wasserstein barycenters is NP-hard even in simple cases, by linking it to a known NP-hard problem and analyzing the complexity of related decision problems.

Contribution

It establishes the NP-hardness of finding sparse Wasserstein barycenters and introduces a related decision problem, SCMP, with complexity implications for barycenter computation.

Findings

Finding sparse barycenters is NP-hard in dimension 2 for three measures.

Verification of transport cost and sparsity for a given measure is polynomial-time.

The encoding size of barycenters can be exponentially larger than that of the measures.

Abstract

The discrete Wasserstein barycenter problem is a minimum-cost mass transport problem for a set of probability measures with finite support. In this paper, we show that finding a barycenter of sparse support is hard, even in dimension 2 and for only 3 measures. We prove this claim by showing that a special case of an intimately related decision problem SCMP -- does there exist a measure with a non-mass-splitting transport cost and support size below prescribed bounds? -- is NP-hard for all rational data. Our proof is based on a reduction from planar 3-dimensional matching and follows a strategy laid out by Spieksma and Woeginger (1996) for a reduction to planar, minimum circumference 3-dimensional matching. While we closely mirror the actual steps of their proof, the arguments themselves differ fundamentally due to the complex nature of the discrete barycenter problem. Containment of SCMP in NP will remain open. We prove that, for a given measure, sparsity and cost of an optimal transport to a set of measures can be verified in polynomial time in the size of a bit encoding of the measure. However, the encoding size of a barycenter may be exponential in the encoding size of the underlying measures.