Parity Polytopes and Binarization

TL;DR

This paper studies generalized parity polytopes with ordering constraints, providing formulations, descriptions, and algorithms, and analyzes their effectiveness in integer programming, especially for the traveling salesman problem.

Contribution

It introduces extended formulations and separation algorithms for generalized parity polytopes with ordering constraints, and analyzes their impact on optimization problems.

Findings

Binarization with parity constraints often does not improve dual bounds.

Extended formulations enable better understanding of these polytopes.

Parity constraints are ineffective for the TSP model studied.

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 $x_i \geq x_{i+1}$ 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.