Scenario reduction revisited: fundamental limits and guarantees
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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.
KeywordsScenario reduction Wasserstein distance Constant-factor approximation algorithm k-median clustering k-means clustering
The authors are indebted to the referees and the guest editors for their comments that considerably improved the manuscript. This research was funded by the SNSF Grant BSCGI0_157733 and the EPSRC Grants EP/M028240/1 and EP/M027856/1.
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