Characterizing the integer points in 2-decomposable polyhedra by closedness under operations
Kei Kimura, Kazuhisa Makino, Shota Yamada, Ryo Yoshizumi
TL;DR
The paper advances a closure-theoretic view of integer points in polyhedra by connecting 2-decomposability to closure under concrete operations. It proves that SVPI, DC, and especially UTVPI representations are precisely captured by specific closure properties: midpoint-neighbor-closed, $\lceil (x+y)/2 \rceil$/$\lfloor (x+y)/2 \rfloor$-closed, and $(\mu, \text{median})$-closed, respectively, while TVPI with infinite inequalities aligns with the convex-closure framework. A key result is that UTVPI representability is equivalent to being closed under $\mu$ and $\text{median}$, bridging discrete convex analysis with polyhedral integer feasibility. The work also characterizes 2-decomposability via weak closure under an infinite family of partial majority operations, clarifying the limitations of finite-arity closure in this context. Together, these results provide a structured, closure-based lens for understanding when integer points arise from specific polyhedral representations, with potential implications for CSPs and discrete convex optimization.
Abstract
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.