Sorting Network Relaxations for Vector Permutation Problems

The Birkhoff polytope (the convex hull of the set of permutation matrices) is frequently invoked informulating relaxations of optimization problems over permutations. The Birkhoff polytope is representedusing Θ(n) variables and constraints, significantly more than the n variables one could use to representa permutation as a vector. Using a recent construction of Goemans [1], we show that when optimizingover the convex hull of the permutation vectors (the permutahedron), we can reduce the number ofvariables and constraints to Θ(n logn) in theory and Θ(n log n) in practice. We modify the recent convexformulation of the 2-SUM problem introduced by Fogel et al. [2] to use this polytope, and demonstratehow we can attain results of similar quality in significantly less computational time for large n. To ourknowledge, this is the first usage of Goemans’ compact formulation of the permutahedron in a convexoptimization problem. We also introduce a simpler regularization scheme for this convex formulation ofthe 2-SUM problem that yields good empirical results.