Characterizing the solution sets in a problem by closedness under operations is recognized as one of the key aspects of algorithm development, especially in constraint satisfaction. An example from the Boolean satisfiability problem is that the solution set of a Horn conjunctive normal form (CNF) is closed under the minimum operation, and this property implies that minimizing a nonnegative linear function over a Horn CNF can be done in polynomial time. In this paper, we focus on the set of integer points (vectors) in a polyhedron, and study the relation between these sets and closedness under operations from the viewpoint of 2-decomposability. By adding further conditions to the 2-decomposable polyhedra, we show that important classes of sets of integer vectors in polyhedra are characterized by 2-decomposability and closedness under certain operations, and in some classes, by closedness under operations alone. The most prominent result we show is that the set of integer vectors in a unit-two-variable-per-inequality polyhedron can be characterized by closedness under the median and directed discrete midpoint operations, each of these operations was independently considered in constraint satisfaction and discrete convex analysis.

翻译：在问题中通过运算封闭性刻画解集被认为是算法发展的关键方面之一，尤其在约束满足领域。布尔可满足性问题的一个例子是：霍恩合取范式（CNF）的解集在最小值运算下封闭，这一性质意味着在霍恩CNF上最小化非负线性函数可在多项式时间内完成。本文聚焦于多面体中的整数点（向量）集合，从2-可分性的视角研究这些集合与运算封闭性之间的关系。通过为2-可分多面体添加额外条件，我们证明多面体中整数向量集合的重要类别可由2-可分性及特定运算封闭性共同刻画，而某些类别仅凭运算封闭性即可刻画。我们展示的最显著结果是：单位双变量不等式多面体中的整数向量集合可由中位数运算和定向离散中点运算封闭性刻画，这两种运算此前分别在约束满足与离散凸分析中被独立研究过。