State of the Art in Global Optimization pp 57-73 | Cite as

# A Branch and Bound Algorithm for the Quadratic Assignment Problem using a Lower Bound Based on Linear Programming

## Abstract

In this paper, we study a branch and bound algorithm for the quadratic assignment problem (QAP) that uses a lower bound based on the linear programming (LP) relaxation of a classical integer programming formulation of the QAP. We report on computational experience with the branch and bound algorithm on all QAP test problems of dimension *n* ≤ 15 of QAPLIB, a standard library of QAP test problems. The linear programming relaxations are solved with an implementation of an interior point algorithm that uses a preconditioned conjugate gradient algorithm to compute directions. The branch and bound algorithm is compared with a similar branch and bound algorithm that uses the Gilmore-Lawler lower bound (GLB) instead of the LP-based bound. The LP-based algorithm examines a small portion of the nodes explored by the GLB-based algorithm. Extensions to the implementation are discussed.

## Keywords

Branch and Bound Combinatorial Optimization Interior Point Methods Linear Programming Lower Bounds Quadratic Assignment Problem## Preview

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