# Minimum energy configurations on a toric lattice as a quadratic assignment problem

## TL;DR

This paper compares bounds for the quadratic assignment problem and applies the strongest to optimize particle configurations on a toric grid, proving some configurations are energy-minimizing.

## Contribution

It proves the SDP bound is stronger than the CQP bound and applies symmetry reduction to compute bounds for energy minimization on a toric lattice.

## Key findings

- SDP bound is stronger than CQP bound.
- Optimality of certain particle configurations is established.
- Symmetry reduction enables feasible computation of large SDPs.

## Abstract

We consider three known bounds for the quadratic assignment problem (QAP): an eigenvalue, a convex quadratic programming (CQP), and a semidefinite programming (SDP) bound. Since the last two bounds were not compared directly before, we prove that the SDP bound is stronger than the CQP bound. We then apply these to improve known bounds on a discrete energy minimization problem, reformulated as a QAP, which aims to minimize the potential energy between repulsive particles on a toric grid. Thus we are able to prove optimality for several configurations of particles and grid sizes, complementing earlier results by Bouman, Draisma and Van Leeuwaarden [ SIAM Journal on Discrete Mathematics, 27(3):1295--1312, 2013]. The semidefinite programs in question are too large to solve without pre-processing, and we use a symmetry reduction method by Parrilo and Permenter [Mathematical Programming, 181:51--84, 2020] to make computation of the SDP bounds possible.