Combinatorial Assortment Optimization

  • Nicole Immorlica
  • Brendan Lucier
  • Jieming MaoEmail author
  • Vasilis Syrgkanis
  • Christos Tzamos
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11316)


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 choice model in which each consumer selects a utility-maximizing bundle subject to a private valuation function, and study the complexity of the resulting optimization problem. Our main result is an exact algorithm for k-additive valuations, under a model of vertical differentiation in which customers agree on the relative value of each pair of items but differ in their absolute willingness to pay. For valuations that are vertically differentiated but not necessarily k-additive, we show how to obtain constant approximations under a “well-priced” condition, where each product’s price is sufficiently high. We further show that even for a single customer with known valuation, any sub-polynomial approximation to the problem requires exponentially many demand queries when the valuation function is XOS, and that no FPTAS exists even when the valuation is succinctly representable.



We would like to thank Aviad Rubinstein for pointing out an improvement on Theorem 7.


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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Nicole Immorlica
    • 1
  • Brendan Lucier
    • 1
  • Jieming Mao
    • 2
    Email author
  • Vasilis Syrgkanis
    • 1
  • Christos Tzamos
    • 3
  1. 1.Microsoft ResearchBostonUSA
  2. 2.University of PennsylvaniaPhiladelphiaUSA
  3. 3.University of Wisconsin-MadisonMadisonUSA

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