Parity polytopes and binarization

We consider generalizations of parity polytopes whose variables, in addition to a parity constraint, satisfy certain ordering constraints. More precisely, the variable domain is partitioned into k contiguous groups, and within each group, we require x_i≥x_i+1 for all relevant i. Such constraints are used to break symmetry after replacing an integer variable by a sum of binary variables, so-called binarization. We provide extended formulations for such polytopes, derive a complete outer description, and present a separation algorithm for the new constraints. It turns out that applying binarization and only enforcing parity constraints on the new variables is often a bad idea. For our application, an integer programming model for the graphic traveling salesman problem, we observe that parity constraints do not improve the dual bounds, and we provide a theoretical explanation of this effect.