How to start a heuristic? Utilizing lower bounds for solving the quadratic assignment problem

Radomil Matousek,Jakub Kudela,Ladislav Dobrovsky

doi:10.5267/j.ijiec.2021.12.003

Radomil Matousek, Jakub Kudela + Show 1 more

Open Access

https://doi.org/10.5267/j.ijiec.2021.12.003

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Abstract

The Quadratic Assignment Problem (QAP) is one of the classical combinatorial optimization problems and is known for its diverse applications. The QAP is an NP-hard optimization problem which attracts the use of heuristic or metaheuristic algorithms that can find quality solutions in an acceptable computation time. On the other hand, there is quite a broad spectrum of mathematical programming techniques that were developed for finding the lower bounds for the QAP. This paper presents a fusion of the two approaches whereby the solutions from the computations of the lower bounds are used as the starting points for a metaheuristic, called HC12, which is implemented on a GPU CUDA platform. We perform extensive computational experiments that demonstrate that the use of these lower bounding techniques for the construction of the starting points has a significant impact on the quality of the resulting solutions.

Highlights

The NP-hard quadratic assignment problem (QAP) in its Koopmans and Beckmann form (Koopmans & Beckmann, 1957), which is notoriously difficult in practice, can be described as follows (Cela, 2013): The problem is structured on a complete directed graph G = (V, A) with n nodes and n arcs, together with a set of n facilities that have to be assigned to the nodes
This paper presents a fusion of the two approaches whereby the solutions from the computations of the lower bounds are used as the starting points for a metaheuristic, called HC12, which is implemented on a Graphics Processing Unit (GPU) compute unified device architecture (CUDA) platform
The HC12 algorithm was implemented for HPC computations on GPU CUDA 7.x (i.e., NVIDIA RTX 2080, 8GB), where not more than 6GB were used for any of the QAP instances

Summary

Introduction

The NP-hard quadratic assignment problem (QAP) in its Koopmans and Beckmann form (Koopmans & Beckmann, 1957), which is notoriously difficult in practice, can be described as follows (Cela, 2013): The problem is structured on a complete directed graph G = (V, A) with n nodes and n arcs, together with a set of n facilities that have to be assigned to the nodes. By using binary variables x , = 1 if facility f is assigned to node i, and 0 otherwise, the QAP can be stated as the following quadratic 0-1 optimization problem: min a , b, x x, (1) ∈ ∈∈ ∈ s.t. x , = 1 ∀f ∈ N (2) ∈. It is quite well known that the constraint matrix, defined by (2)-(3) is totally unimodular, implying that the optimization of any linear objective function over the QAP feasible set is just a relatively easy linear programming problem, known as the linear assignment problem (LAP) (Burkard et al, 2012).

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