# Scenario Reduction Revisited: Fundamental Limits and Guarantees

arXiv: https://arxiv.org/abs/1701.04072 · HTML: https://mlsanity.com/paper/1701.04072/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Jan 2017.

## Built on by (most cited first, 4 of 4)

- [Low Budget Active Learning via Wasserstein Distance: An Integer Programming Approach](https://mlsanity.com/paper/2106.02968/) (2021)
- [Semi-Discrete Optimal Transport: Hardness, Regularization and Numerical Solution](https://mlsanity.com/paper/2103.06263/) (2021)
- [HGCN2SP: Hierarchical Graph Convolutional Network for Two-Stage Stochastic Programming](https://mlsanity.com/paper/2511.16027/) (2025)
- [Uncertainty Informed Optimal Resource Allocation with Gaussian Process based Bayesian Inference](https://mlsanity.com/paper/2307.00032/) (2023)

## Abstract (arXiv)

The goal of scenario reduction is to approximate a given discrete distribution with another discrete distribution that has fewer atoms. We distinguish continuous scenario reduction, where the new atoms may be chosen freely, and discrete scenario reduction, where the new atoms must be chosen from among the existing ones. Using the Wasserstein distance as measure of proximity between distributions, we identify those n-point distributions on the unit ball that are least susceptible to scenario reduction, i.e., that have maximum Wasserstein distance to their closest m-point distributions for some prescribed m<n. We also provide sharp bounds on the added benefit of continuous over discrete scenario reduction. Finally, to our best knowledge, we propose the first polynomial-time constant-factor approximations for both discrete and continuous scenario reduction as well as the first exact exponential-time algorithms for continuous scenario reduction.