Sorting Network Relaxations for Vector Permutation Problems

TL;DR

This paper introduces a compact convex relaxation for permutation problems using a new polytope representation, reducing computational complexity and enabling efficient solutions for large-scale instances.

Contribution

It applies Goemans' permutahedron formulation to convex relaxations, significantly reducing variables and constraints in permutation optimization problems.

Findings

Reduced variables from Θ(n^2) to Θ(n log n)

Achieved similar solution quality with less computational time

Introduced a new regularization scheme for the convex formulation

Abstract

The Birkhoff polytope (the convex hull of the set of permutation matrices) is frequently invoked in formulating relaxations of optimization problems over permutations. The Birkhoff polytope is represented using $\Theta(n^2)$ variables and constraints, significantly more than the $n$ variables one could use to represent a permutation as a vector. Using a recent construction of Goemans (2010), 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 $\Theta(n \log n)$ in theory and $\Theta(n \log^2 n)$ in practice. We modify the recent convex formulation of the 2-SUM problem introduced by Fogel et al. (2013) 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.