We advance the theoretical study of $\{0, 1/2\}$-cuts for integer programming problems $\max\{c^T x \colon A x \leq b, x \text{ integer}\}$. Such cuts are Gomory-Chv\'atal cuts that only need multipliers of value $0$ or $1/2$ in their derivation. The intersection of all $\{0, 1/2\}$-cuts derived from $Ax \le b$ is denoted by $P_{1/2}$ and called the $\{0,1/2\}$-closure of $P = \{x : Ax \le b\}$. The primal separation problem for $\{0, 1/2\}$-cuts is: Given a vertex $\hat x$ of the integer hull of $P$ and some fractional point $x^* \in P$, does there exist a $\{0,1/2\}$-cut that is tight at $\hat x$ and violated by $x^*$? Primal separation is the key ingredient of primal cutting-plane approaches to integer programming. In general, primal separation for $\{0,1/2\}$-cuts is NP-hard. We present two cases for which primal separation is solvable in polynomial time. As an interesting side product, we obtain a(nother) simple proof that matching can be solved in polynomial time. Furthermore, since optimization over the Gomory-Chv\'atal closure is also NP-hard, there has been recent research on solving the optimization problem over the Gomory-Chv\'atal closure approximately. In a similar spirit, we show that the optimization problem over the $\{0,1/2\}$-closure can be solved in polynomial time up to a factor $(1 + \varepsilon)$, for any fixed $\varepsilon > 0$.

翻译：我们推进了整数规划问题 $\max\{c^T x \colon A x \leq b, x \text{ 整数}\}$ 中 $\{0, 1/2\}$-割的理论研究。此类割是Gomory-Chvátal割，其推导过程中仅需乘子取值为 $0$ 或 $1/2$。由 $Ax \le b$ 导出的所有 $\{0, 1/2\}$-割的交集记为 $P_{1/2}$，称为 $P = \{x : Ax \le b\}$ 的 $\{0,1/2\}$-闭包。$\{0, 1/2\}$-割的原分离问题可表述为：给定 $P$ 的整数壳的一个顶点 $\hat x$ 及某个分数点 $x^* \in P$，是否存在一个在 $\hat x$ 处取紧且被 $x^*$ 违反的 $\{0,1/2\}$-割？原分离是整数规划中原切割平面方法的关键要素。一般而言，$\{0,1/2\}$-割的原分离是NP难的。我们给出了两个可在多项式时间内求解原分离的案例。作为有趣的副产品，我们（再次）得到了一个关于匹配问题可在多项式时间内求解的简洁证明。此外，由于在Gomory-Chvátal闭包上的优化同样是NP难的，近期研究致力于近似求解该优化问题。类似地，我们证明对于任意固定 $\varepsilon > 0$，$\{0,1/2\}$-闭包上的优化问题可在 $(1 + \varepsilon)$ 近似因子内于多项式时间内求解。