可由UTVPI表示的整数点集：离散凸性、多态性与成对闭包
UTVPI-representable integer point sets: discrete convexity, polymorphisms, and pairwise closure
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中文总结 AI 辅助
该研究关联整数点集的不等式表示、离散凸性等五个视角，完全刻画了UTVPI可表示性，建立成对闭包理论并确定相关类的包含层次。
中文摘要 AI 辅助
我们研究由单变量不等式（SVPI）、差分约束（DC）、单位双变量不等式（UTVPI）及双变量不等式（TVPI）系统表示的整数格点子集，关联五个视角：不等式表示、离散凸性、多态性、从两坐标投影的重构以及闭包算子的不动点。核心结果完全刻画了UTVPI可表示性：对每个满足n>1的集合S⊆ℤⁿ，S是UTVPI可表示的当且仅当S在定向中点运算和中位数运算下封闭，当且仅当S是整数凸且2-可分解的。中位数条件可替换为在某个多数运算下封闭，该类也是成对定向中点闭包算子的不动点类，因此五个视角对UTVPI可表示性给出等价刻画，其中2-可分解性是将已知的二维整数凸性与UTVPI可表示性的等价性提升至任意维度所需的全局条件。该定理嵌入更广泛的成对闭包理论：对运算族F，我们通过将每个两坐标投影在F下封闭并合并所得集合定义闭包算子，其不动点恰是同时为2-可分解且F-封闭的集合，我们为此类刻画建立了局部到全局准则。闭凸包类似物刻画了TVPI可表示性，我们还通过自然多运算刻画SVPI可表示性，证明了若干相关类的基于运算的刻画的局限性，并确定了一般、布尔及二维场景下的完整包含层次。
英文摘要
We study subsets of the integer lattice represented by single-variable-per-inequality (SVPI), difference-constraint (DC), unit two-variable-per-inequality (UTVPI), and two-variable-per-inequality (TVPI) systems. We relate five viewpoints: inequality representation, discrete convexity, polymorphisms, reconstruction from two-coordinate projections, and fixed points of closure operators. Our central result completely characterizes UTVPI-representability. For every set $S\subseteq\mathbb Z^n$ with $n>1$, \[ \begin{aligned} &S\text{ is UTVPI-representable}\\ &\;\Longleftrightarrow\; S\text{ is closed under the directed midpoint and median operations}\\ &\;\Longleftrightarrow\; S\text{ is integrally convex and $2$-decomposable}. \end{aligned} \] The median condition may instead be replaced by closedness under some majority operation, and the same class is the fixed-point class of a pairwise directed-midpoint closure operator. Thus, all five viewpoints yield equivalent characterizations of UTVPI-representability. In particular, $2$-decomposability is exactly the global condition needed to lift the known two-dimensional equivalence between integral convexity and UTVPI-representability to arbitrary dimension. This theorem is embedded in a broader pairwise-closure theory. For a family $F$ of operations, we define a closure operator by closing every two-coordinate projection under $F$ and joining the resulting sets. Its fixed points are precisely the sets that are both $2$-decomposable and $F$-closed, and we establish a local-to-global criterion for such characterizations. A closed-convex-hull analogue characterizes TVPI-representability. We also characterize SVPI-representability by natural multioperations, prove limitations of operation-based characterizations for several related classes, and determine the complete inclusion hierarchies in the general, Boolean, and two-dimensional settings.
发表机构
- Kyushu University（九州大学）
- Kyoto University（京都大学）
- National Institute of Advanced Industrial Science and Technology（产业技术综合研究所）
- NTT Social Informatics Laboratories（NTT社会信息系统实验室）
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