Primal Separation and Approximation for the $\{0, 1/2\}$-closure

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 $\varepsilon > 0$.