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 (Eur J Oper Res 91:611–618, 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.

离散的Wasserstein重心问题是具有有限支持的一组概率测度的最小成本质量运输问题。在本文中，我们表明，即使在2维且仅3个度量中，也很难找到稀疏支持的重心。我们通过证明一个密切相关的决策问题SCMP的特殊情况来证明这一主张-是否存在一种措施，其运输成本和支持规模在规定范围之内？对于所有有理数据而言，NP都是困难的。我们的证明是基于减少平面3维匹配，并遵循Spieksma和Woeginger（Eur J Oper Res 91：611–618，1996）提出的减少平面最小圆周3维匹配的策略。尽管我们仔细反映了他们证明的实际步骤，由于离散重心问题的复杂性，论证本身在根本上有所不同。NP中的SCMP收容将保持开放状态。我们证明，对于给定的度量，可以在多项式时间内以度量的位编码大小来验证对一组度量的最优传输的稀疏性和成本。但是，重心的编码大小在基础度量的编码大小中可能是指数级的。