On Mixing Inequalities: Rank, Closure, and Cutting-Plane Proofs

We study the mixing inequalities that were introduced by Gunluk and Pochet [Math. Program., 90 (2001), pp. 429-457]. We show that a mixing inequality which mixes $n$ MIR inequalities has MIR rank at most $n$ if it is a type I mixing inequality and at most $n-1$ if it is a type II mixing inequality. We also show that these bounds are tight for $n=2$. Given a mixed-integer set $P_I=P\cap Z(I)$, where $P$ is a polyhedron and $Z(I)=\{x\in\mathbb{R}^n\,:\,x_i\in\mathbb{Z}$ $\forall i \in I\}$, we define mixing inequalities for $P_I$. We show that the elementary mixing closure of $P$ with respect to $I$ can be described using a bounded number of mixing inequalities, each of which has a bounded number of terms. This implies that the elementary mixing closure of $P$ is a polyhedron. Finally, we show that any mixing inequality can be derived via a polynomial length MIR cutting-plane proof. Combined with results of Dash [On the complexity of cutting plane proofs using split cuts, IBM Research Report RC 24082, Oct. 2006] and Pudlak [J. Symbolic Logic, 62 (1997), pp. 981-998], this implies that there are valid inequalities for a certain mixed-integer set that cannot be obtained via a polynomial-size mixing cutting-plane proof.