The Largest Unsolved QAP Instance Tai256c Can Be Converted into A 256-dimensional Simple BQOP with A Single Cardinality Constraint

TL;DR

This paper transforms the large, unsolved Tai256c quadratic assignment problem into a simpler 256-dimensional binary quadratic optimization problem with a single constraint, and proposes an improved branch and bound method to tighten the lower bound.

Contribution

It introduces a novel conversion of Tai256c into a simpler BQOP with symmetry properties and develops an efficient branch and bound approach to improve lower bounds.

Findings

Achieved a new lower bound with a 1.36% gap.

Successfully converted Tai256c into a simpler BQOP.

Demonstrated an effective branch and bound method.

Abstract

Tai256c is the largest unsolved quadratic assignment problem (QAP) instance in QAPLIB; a 1.48\% gap remains between the best known feasible objective value and lower bound of the unknown optimal value. This paper shows that the instance can be converted into a 256 dimensional binary quadratic optimization problem (BQOP) with a single cardinality constraint which requires the sum of the binary variables to be 92. The converted BQOP is much simpler than the original QAP tai256c and it also inherits some of the symmetry properties. However, it is still very difficult to solve. We present an efficient branch and bound method for improving the lower bound effectively. A new lower bound with 1.36\% gap is also provided.