Beyond the Birkhoff Polytope: Convex Relaxations for Vector Permutation Problems

The Birkhoff polytope (the convex hull of the set of permutation matrices), which is represented using Θ(n) variables and constraints, is frequently invoked in formulating relaxations of optimization problems over permutations. Using a recent construction of Goemans [1], we show that when optimizing over the convex hull of the permutation vectors (the permutahedron), we can reduce the number of variables and constraints to Θ(n log n) in theory and Θ(n log n) in practice. We modify the recent convex formulation of the 2-SUM problem introduced by Fogel et al. [2] to use this polytope, and demonstrate how we can attain results of similar quality in significantly less computational time for large n. To our knowledge, this is the first usage of Goemans’ compact formulation of the permutahedron in a convex optimization problem. We also introduce a simpler regularization scheme for this convex formulation of the 2-SUM problem that yields good empirical results.