Sinkhorn Algorithm for Lifted Assignment Problems

TL;DR

This paper extends Sinkhorn's algorithm to efficiently solve lifted linear relaxations of the Quadratic Assignment Problem, improving scalability, accuracy, and numerical stability over traditional methods.

Contribution

It introduces a novel Bregman projection algorithm onto the Johnson Adams polytope for lifted QAP relaxations, enhancing scalability and stability.

Findings

The new algorithm is more scalable than standard solvers.

It achieves higher accuracy and numerical stability.

It can solve larger linear programs efficiently.

Abstract

Recently, Sinkhorn's algorithm was applied for approximately solving linear programs emerging from optimal transport very efficiently. This was accomplished by formulating a regularized version of the linear program as Bregman projection problem onto the polytope of doubly-stochastic matrices, and then computing the projection using the efficient Sinkhorn algorithm, which is based on alternating closed-form Bregman projections on the larger polytopes of row-stochastic and column-stochastic matrices. In this paper we suggest a generalization of this algorithm for solving a well-known lifted linear program relaxations of the Quadratic Assignment Problem (QAP), which is known as the Johnson Adams (JA) Relaxation. First, an efficient algorithm for Bregman projection onto the JA polytope by alternating closed-form Bregman projections onto one-sided local polytopes is devised. The one-sided polytopes can be seen as a high-dimensional, generalized version of the row/column-stochastic polytopes. Second, a new method for solving the original linear programs using the Bregman projections onto the JA polytope is developed and shown to be more accurate and numerically stable than the standard approach of driving the regularizer to zero. The resulting algorithm is considerably more scalable than standard linear solvers and is able to solve significantly larger linear programs.