Solving Combinatorial Pricing Problems using Embedded Dynamic Programming Models

2024-03-19

arXiv.org

AI Summary

This paper addresses the Combinatorial Pricing Problem (CPP), a canonical combinatorial bilevel programming problem where a leader sets item tolls to maximize revenue, while a follower solves a combinatorial optimization subproblem subject to the induced cost constraints. To overcome the scalability limitations of conventional value-function-based approaches, we propose the first single-level dual formulation that embeds CPP into a dynamic programming framework. Specifically, we reformulate the follower’s problem as a longest-path problem on a directed acyclic graph and introduce a novel “Selection Diagram” structure—a compact decision diagram encoding feasible follower choices. We further pioneer the integration of decision diagrams with cutting-plane methods for efficient solution. Our approach significantly outperforms state-of-the-art algorithms on three CPP variants and the knapsack interdiction problem, substantially expanding the scale of combinatorial bilevel programs solvable to provable optimality.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Constraint Programming
• Planning, Routing, and Scheduling: Mixed Discrete/Continuous Planning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

Abstract

The combinatorial pricing problem (CPP) is a bilevel problem in which the leader maximizes their revenue by imposing tolls on certain items that they can control. Based on the tolls set by the leader, the follower selects a subset of items corresponding to an optimal solution of a combinatorial optimization problem. To accomplish the leader's goal, the tolls need to be sufficiently low to discourage the follower from choosing the items offered by the competitors. In this paper, we derive a single-level reformulation for the CPP by rewriting the follower's problem as a longest path problem using a dynamic programming model, and then taking its dual and applying strong duality. We proceed to solve the reformulation in a dynamic fashion with a cutting plane method. We apply this methodology to two distinct dynamic programming models, namely, a novel formulation designated as selection diagram and the well-known decision diagram. We also produce numerical results to evaluate their performances across three different specializations of the CPP and a closely related problem that is the knapsack interdiction problem. Our results showcase the potential of the two proposed reformulations over the natural value function approach, expanding the set of tools to solve combinatorial bilevel programs.

Problem

• Solving bilevel combinatorial pricing via dynamic programming
• Reformulating follower's problem as longest path using duality
• Evaluating performance on CPP and knapsack interdiction problems

Innovation

• Reformulate CPP as single-level using dynamic programming
• Apply cutting plane method dynamically for solution
• Utilize selection and decision diagrams for modeling