Sequential Submodular Maximization and Applications to Ranking an Assortment of Products

49 Pages Posted: 18 Mar 2020 Last revised: 25 Jan 2022

See all articles by Arash Asadpour

Arash Asadpour

City University of New York

Rad Niazadeh

University of Chicago - Booth School of Business

Amin Saberi

Stanford University - Department of Management Science & Engineering

Ali Shameli

Stanford University, Management Science & Engineering

Date Written: February 21, 2020

Abstract

We study a submodular maximization problem motivated by applications in online retail. A platform displays a list of products to a user in response to a search query. The user inspects the first k items in the list for a k chosen at random from a given distribution, and decides whether to purchase an item from that set based on a choice model. The goal of the platform is to maximize the engagement of the shopper defined as the probability of purchase. This problem gives rise to a less-studied variation of submodular maximization in which we are asked to choose an ordering of a set of elements to maximize a linear combination of different submodular functions.

First, using a reduction to maximizing submodular functions over matroids, we give an optimal (1-1/e)-approximation for this problem. We then consider a variant in which the platform cares not only about user engagement, but also about diversification across various groups of users, that is, guaranteeing a certain probability of purchase in each group. We characterize the polytope of feasible solutions and give a bi-criteria ((1-1/e)^2,(1-1/e)^2)-approximation for this problem by rounding an approximate solution of a linear programming relaxation. For rounding, we relay on our reduction and the particular rounding techniques for matroid polytopes. For the special case in which underlying submodular functions are coverage functions -- which is practically relevant in online retail -- we propose an alternative LP relaxation and a simpler randomized rounding for the problem. This approach yields to an optimal bi-criteria (1-1/e,1-1/e)-approximation algorithm for the special case of the problem with coverage functions.

Keywords: Product Ranking, Submodular Maximization, Online Platforms, Online Retail, Group Fairness

Suggested Citation

Asadpour, Arash and Niazadeh, Rad and Saberi, Amin and Shameli, Ali, Sequential Submodular Maximization and Applications to Ranking an Assortment of Products (February 21, 2020). Chicago Booth Research Paper No. 20-26, Available at SSRN: https://ssrn.com/abstract=3542382 or http://dx.doi.org/10.2139/ssrn.3542382

Arash Asadpour

City University of New York ( email )

55 Lexington Ave
New York, NY NY 10010
United States

Rad Niazadeh (Contact Author)

University of Chicago - Booth School of Business ( email )

5807 S Woodlawn Ave
Chicago, IL 60637

HOME PAGE: http://radniazadeh.github.io/

Amin Saberi

Stanford University - Department of Management Science & Engineering ( email )

473 Via Ortega
Stanford, CA 94305-9025
United States

Ali Shameli

Stanford University, Management Science & Engineering ( email )

473 Via Ortega
Stanford, CA 94305-9025
United States

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