We introduce the integrality number of an integer program (IP). Roughly speaking, the integrality number is the smallest number of integer constraints needed to solve an IP via a mixed integer (MIP) relaxation. One notable property of this number is its invariance under unimodular transformations of the constraint matrix. Considering the largest minor \(\varDelta \) of the constraint matrix, our analysis allows us to make statements of the following form: there exists a number \(\tau (\varDelta )\) such that an IP with n many variables and \(n + \sqrt{n /\tau (\varDelta )}\) many inequality constraints can be solved via a MIP relaxation with fewer than n integer constraints. From our results it follows that IPs defined by only n constraints can be solved via a MIP relaxation with \(O(\sqrt{\varDelta })\) many integer constraints.

我们介绍一个整数程序（IP）的完整性编号。粗略地说，完整性数是通过混合整数（MIP）松弛来求解IP所需的最小整数约束。此数字的一个显着特性是在约束矩阵的单模变换下其不变性。考虑到约束矩阵的最大次要\（\ varDelta \），我们的分析使我们能够做出以下形式的陈述：存在一个数字\（\ tau（\ varDelta）\），这样一个IP具有n个许多变量，并且\（n + \ sqrt {n / \ tau（\ varDelta）} \）通过不超过n的MIP松弛可以解决许多不等式约束整数约束。根据我们的结果，可以得出结论，仅通过n个约束条件定义的IP可以通过MIP松弛并使用\（O（\ sqrt {\ varDelta}）\）许多整数约束条件来求解。