In this paper, we study the problem of computing the lattice point closure of a conjunction of Unit Two Variable Per Inequality (UTVPI) constraints. We accomplish this by adapting Johnson’s all pairs shortest path algorithm to UTVPI constraint systems (UCSs). Thus, we obtain a closure algorithm that is efficient for sparse constraint systems. This problem has been extremely well-studied in the literature, since it arises in a number of applications, including but not limited to, program verification and operations research. In UTVPI constraints, linear feasibility does not always imply integer feasibility. Thus, there is a difference between the linear closure of a UCS and the integer closure of that same system. Finding the linear closure requires only a single inference rule called the transitive inference rule. This inference rule corresponds to the addition of constraints and preserves both linear and integer solutions. The problem of finding the integer closure requires the use of the tightening inference rule. Unlike the transitive inference rule, the tightening inference rule does not preserve linear solutions. However, it does preserve integer solutions. The complexity of solving the integer closure problem has steadily improved over the past several decades with the fastest algorithm for this problem running in time O(n3) on a UCS with n variables and m constraints. For the same input parameters, we detail an algorithm that runs in time $O(m\cdot n +n^{2} \cdot \log n)$ . It is clear that our algorithm is superior to the state of the art when the UCS is sparse (m ∈ o(n2)), and no worse than the state of the art when the UCS is dense (m ∈Θ(n2)). The best known running time for computing the closure of a conjunction of difference constraints (m constraints, n variables) is $O(m\cdot n +n^{2} \cdot \log n)$ , and UTVPI constraints subsume difference constraints.

中文翻译：

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关于每个不等式约束的单位二变量系统中的整数闭包

在本文中，我们研究了计算单位二变量每不等式 (UTVPI) 约束的合的格点闭包的问题。我们通过将 Johnson 的所有对最短路径算法适应 UTVPI 约束系统 (UCS) 来实现这一点。因此，我们获得了一种对稀疏约束系统有效的闭包算法。这个问题在文献中得到了非常深入的研究，因为它出现在许多应用中，包括但不限于程序验证和运筹学。在 UTVPI 约束中，线性可行性并不总是意味着整数可行性。因此，UCS 的线性闭包与同一​​系统的整数闭包之间存在差异。找到线性闭包只需要一个称为传递推理规则的推理规则。此推理规则对应于约束的添加并保留线性和整数解。找到整数闭包的问题需要使用紧缩推理规则。与传递推理规则不同，紧缩推理规则不保留线性解。但是，它确实保留了整数解。在过去的几十年中，解决整数闭包问题的复杂性稳步提高，该问题的最快算法在具有 n 个变量和 m 个约束的 UCS 上运行时间为 O(n3)。对于相同的输入参数，我们详细介绍了一个在时间 $O(m\cdot n +n^{2} \cdot \log n)$ 中运行的算法。很明显，当 UCS 稀疏时（m ∈ o(n2)），我们的算法优于最先进的算法，并且不比 UCS 稠密时的最先进算法（m ∈Θ(n2)）差.