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Submodular Order Functions and Assortment Optimization

Rajan Udwani

TL;DR

This work introduces submodular order functions, a novel structural framework that imposes a limited form of submodularity relative to a permutation of the ground set. The authors develop threshold-based augmentation and directed local-search algorithms that, under cardinality, budget, and matroid constraints, achieve tight or near-tight constant-factor approximations, with explicit query complexities. They apply the framework to constrained assortment optimization across MNL, MMNL, and Markov choice models, deriving improved guarantees and, in some cases, constant-factor performance for joint customization and pricing. A key contribution is the connection to streaming submodular maximization: functions with a known submodular order yield one-pass, low-memory algorithms that recover the best-known streaming guarantees. The work also establishes a fundamental 0.5 upper bound on polynomial-query algorithms for submodular-order objectives, and it provides a comprehensive framework for extending these ideas to compatible choice models and beyond.

Abstract

We define a new class of set functions that in addition to being monotone and subadditive, also admit a very limited form of submodularity defined over a permutation of the ground set. We refer to this permutation as a submodular order. This class of functions includes monotone submodular functions as a sub-family. We give fast algorithms with strong approximation guarantees for maximizing submodular order functions under a variety of constraints and show a nearly tight upper bound on the highest approximation guarantee achievable by algorithms with polynomial query complexity. Applying this new notion to the problem of constrained assortment optimization in fundamental choice models, we obtain new algorithms that are both faster and have stronger approximation guarantees (in some cases, first algorithm with constant factor guarantee). We also show an intriguing connection to the maximization of monotone submodular functions in the streaming model, where we recover best known approximation guarantees as a corollary of our results.

Submodular Order Functions and Assortment Optimization

TL;DR

This work introduces submodular order functions, a novel structural framework that imposes a limited form of submodularity relative to a permutation of the ground set. The authors develop threshold-based augmentation and directed local-search algorithms that, under cardinality, budget, and matroid constraints, achieve tight or near-tight constant-factor approximations, with explicit query complexities. They apply the framework to constrained assortment optimization across MNL, MMNL, and Markov choice models, deriving improved guarantees and, in some cases, constant-factor performance for joint customization and pricing. A key contribution is the connection to streaming submodular maximization: functions with a known submodular order yield one-pass, low-memory algorithms that recover the best-known streaming guarantees. The work also establishes a fundamental 0.5 upper bound on polynomial-query algorithms for submodular-order objectives, and it provides a comprehensive framework for extending these ideas to compatible choice models and beyond.

Abstract

We define a new class of set functions that in addition to being monotone and subadditive, also admit a very limited form of submodularity defined over a permutation of the ground set. We refer to this permutation as a submodular order. This class of functions includes monotone submodular functions as a sub-family. We give fast algorithms with strong approximation guarantees for maximizing submodular order functions under a variety of constraints and show a nearly tight upper bound on the highest approximation guarantee achievable by algorithms with polynomial query complexity. Applying this new notion to the problem of constrained assortment optimization in fundamental choice models, we obtain new algorithms that are both faster and have stronger approximation guarantees (in some cases, first algorithm with constant factor guarantee). We also show an intriguing connection to the maximization of monotone submodular functions in the streaming model, where we recover best known approximation guarantees as a corollary of our results.

Paper Structure

This paper contains 29 sections, 29 theorems, 114 equations, 2 tables, 9 algorithms.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Applications of Submodular Order Functions
  4. Constrained Assortment Optimization
  5. New Results for Assortment Optimization
  6. Results for MNL and MMNL with Customization.
  7. Results for Markov Model and Beyond.
  8. Application to Streaming Submodular Maximization
  9. Algorithms for Submodular Order Functions
  10. Threshold Based Augmentation in Submodular Order
  11. Local Search Along Submodular Order
  12. Algorithms for Constrained Assortment Optimization
  13. Approximation Guarantees
  14. Cardinality Constraint
  15. Budget Constraint
  16. ...and 14 more sections

Key Result

Theorem 1

Any algorithm that makes polynomial number of queries (in size $n$ of ground set), cannot have approximation guarantee better than $n^{-\Omega(1)}$ for maximizing a monotone subadditive function subject to cardinality constraint.

Theorems & Definitions (53)

  • Theorem 1: Adapted from Theorem 6.1 in mor
  • Theorem 2
  • Theorem 3: Upper Bound
  • Theorem 4
  • Theorem 5
  • Remark 1
  • Theorem 6
  • Theorem 7
  • Corollary 1
  • Theorem 8
  • ...and 43 more