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.

翻译：组合定价问题（CPP）是一个双层优化问题，其中领导者通过对自身可控的特定项目施加收费来实现收益最大化。在领导者设定收费的基础上，跟随者通过求解一个组合优化问题的最优解来选择相应的项目子集。为实现领导者目标，收费需设定得足够低，以阻止跟随者选择竞争对手提供的项目。本文通过将跟随者问题重构为基于动态规划模型的最长路径问题，进而对其取对偶并应用强对偶定理，推导出CPP的单层重构形式。我们采用割平面法以动态方式求解该重构问题。将此方法应用于两种不同的动态规划模型：一种称为选择图的新型建模方法，以及众所周知的决策图模型。我们通过数值实验评估了它们在CPP的三种不同特例及其紧密相关问题——背包拦截问题中的性能表现。研究结果证明，相较于传统的价值函数方法，两种提出的重构模型具有显著优势，从而拓展了解组合双层规划问题的工具集。