# Scenario Reduction Revisited: Fundamental Limits and Guarantees

## TL;DR

This paper investigates the fundamental limits of scenario reduction using Wasserstein distance, identifying distributions most resistant to reduction, and introduces new approximation algorithms with guarantees for both discrete and continuous cases.

## Contribution

It provides the first polynomial-time constant-factor approximation algorithms and exact exponential-time algorithms for scenario reduction under Wasserstein distance.

## Key findings

- Identifies distributions with maximum Wasserstein distance to their reduced forms.
- Provides sharp bounds on the benefits of continuous over discrete scenario reduction.
- Introduces the first polynomial-time approximation algorithms and exact algorithms for scenario reduction.

## Abstract

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.