Parity Polytopes and Binarization

Abstract

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 that $x_i\ge x_{i+1}$ 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.

9 pages, 1 figure, presented at 15th Cologne-Twente Workshop on Graphs and Combinatorial Optimization 2017