We propose a new combinatorial optimization problem that we call the submodular reassignment problem. We are given $k$ submodular functions over the same ground set, and we want to find a set that minimizes the sum of the distances to the sets of minimizers of all functions. The problem is motivated by a two-stage stochastic optimization problem with recourse summarized as follows. We are given two tasks to be processed and want to assign a set of workers to maximize the sum of profits. However, we do not know the value functions exactly, but only know a finite number of possible scenarios. Our goal is to determine the first-stage allocation of workers to minimize the expected number of reallocated workers after a scenario is realized at the second stage. This problem can be modeled by the submodular reassignment problem. We prove that the submodular reassignment problem can be solved in strongly polynomial time via submodular function minimization. We further provide a maximum-flow formulation of the problem that enables us to solve the problem without using a general submodular function minimization algorithm, and more efficiently both in theory and in practice. In our algorithm, we make use of Birkhoff’s representation theorem for distributive lattices.

我们提出了一个新的组合优化问题，称为亚模块重分配问题。我们得到了$ķ$在同一地面集合上的亚模函数，我们希望找到一个集合，以使到所有函数的极小值集合的距离之和最小。该问题是由两阶段的随机优化问题引起的，其追索权归纳如下。我们有两个待处理的任务，并且想要分配一组工人以使利润之和最大化。但是，我们并不确切知道值函数，而只知道有限数量的可能情况。我们的目标是确定第二阶段的情况后，确定第一阶段的工作人员分配，以最大程度地减少预期的重新分配工人数。这个问题可以用亚模重新分配问题来建模。我们证明了通过子模函数最小化可以在强多项式时间内解决子模重分配问题。我们进一步提供了问题的最大流公式化，使我们能够在不使用常规子模块函数最小化算法的情况下解决问题，并且在理论上和实践上都更加有效。在我们的算法中，我们将Birkhoff表示定理用于分布格。