Combinatorial Assortment Optimization

TL;DR

This paper studies the complex problem of designing product assortments where customers can buy bundles, introducing new algorithms and complexity results for various valuation models and customer behaviors.

Contribution

It introduces the combinatorial assortment problem, analyzes its computational complexity, and provides algorithms and approximation results for specific valuation classes.

Findings

Exponential demand queries are needed for XOS valuations.

No FPTAS exists for submodular valuations.

Constant approximations are possible under certain pricing conditions.

Abstract

Assortment optimization refers to the problem of designing a slate of products to offer potential customers, such as stocking the shelves in a convenience store. The price of each product is fixed in advance, and a probabilistic choice function describes which product a customer will choose from any given subset. We introduce the combinatorial assortment problem, where each customer may select a bundle of products. We consider a model of consumer choice where the relative value of different bundles is described by a valuation function, while individual customers may differ in their absolute willingness to pay, and study the complexity of the resulting optimization problem. We show that any sub-polynomial approximation to the problem requires exponentially many demand queries when the valuation function is XOS, and that no FPTAS exists even for succinctly-representable submodular valuations. On the positive side, we show how to obtain constant approximations under a "well-priced" condition, where each product's price is sufficiently high. We also provide an exact algorithm for $k$-additive valuations, and show how to extend our results to a learning setting where the seller must infer the customers' preferences from their purchasing behavior.