An effective Parallel Multistart Tabu Search for Quadratic Assignment Problem on CUDA platform

Abstract

NVidia’s powerful GPU hardware and CUDA platform enables the design of very fast parallel algorithms. Relatively little research has been done so far on GPU implementations of algorithms for computationally demanding discrete optimisation problems. In this paper, the well-known $\mathcal{NP}$-hard Quadratic Assignment Problem (QAP) is considered. Detailed analysis of parallelisation possibilities, memory organisation and access patterns, enables the implementation of fast and effective heuristics for QAP on the GPU — the Parallel Multistart Tabu Search (PMTS). Computational experiments show that PMTS is capable of providing good quality (often optimal or the best known) solutions in a short time, and even better quality solutions in longer runs. PMTS runs up to $420\times$ faster than a single-core counterpart, or $70\times$ faster than a parallel CPU implementation on a high-end six-core CPU.

Introduction

The Quadratic Assignment Problem (QAP) was first defined by Koopmans and Beckmann  [19]. In QAP $n$ facilities have to be assigned to $n$ locations in order to minimise the cost. Distances between locations are stored in $n\times n$ matrix $d$  and flows between facilities in $n\times n$ matrix $f$. Total cost of any assignment (permutation) $\pi$ can be calculated as:

$$C\left(\pi \right)=\sum _{i=1}^n\sum _{j=1}^n{d}_{i,j}\cdot f_{\pi \left(i\right),\pi \left(j\right)}\text{.}$$

The objective is to find assignment ${\pi }^{\ast }$, which minimises the cost function.

The most popular application for QAP is facility layout, e.g. planning university campus  [9] or hospital facilities  [12], [20], or more recently, DNA microarray layout  [8]. In addition, QAP was used to solve Steinberg wiring problem  [25], [2], to design a typewriter keyboard  [24], and even dartboard design  [11]. A recent survey on QAP history and applications is provided by Loiola et al.  [21].

Unfortunately, QAP belongs to the class of $\mathcal{NP}$-hard problems  [14], therefore it is feasible to compute the exact solution only for relatively small instances, with $n$ values up to 30  [21]. This limitation raised interest in heuristics for finding near-optimum solutions for QAP. They include algorithms based on Tabu Search (e.g.  [28]), Simulated Annealing (e.g.  [5]), Scatter Search (e.g.  [6]), Genetic Algorithms (e.g.  [10]), Memetic Algorithms (e.g.  [23]), Ant Colony Optimisation (e.g.  [13], [27]) or Iterated Local Search (e.g.  [26]). Even with such sophisticated techniques, the time required to obtain good quality solutions varies from several minutes for medium size datasets, to a few hours for the larger ones (e.g.  [17]).

The introduction of CUDA and very powerful GPU hardware designed for High Performance Computing, have created an opportunity to considerably speed up computation. Several papers have been published on techniques to speed up heuristics for discrete optimisation problems. Janiak et al.  [18] proposed Tabu Search for Travelling Salesman and Permutation Flowshop Problems, but using pre-CUDA API (Microsoft XNA). Luong et al.  [22] proposed a GPU implementation of Tabu Search for QAP. The shortcoming of both implementations is that only evaluation is done on the GPU. The remaining operations are performed on the CPU, effectively harming performance by introducing CPU–GPU data transfer after each iteration. CUDA Tabu Search for Permutation Flowshop Problem presented in  [7] alleviates that problem by introducing GPU implementation of the tabu list. Finally, CUDA implementation of Tabu Search for QAP (SIMD-TS), described by Zhu et al.  [29], employs non-cooperative parallelism by running multiple independent Tabu Search instances. Since classic Tabu Search is deterministic, certain randomness is introduced to diversify the search paths.

This paper proposes Parallel Multistart Tabu Search for QAP, which aims to connect the effectiveness of the Multistart Tabu Search approach, and high performance provided by GPU hardware. It employs the GPU implementation of the tabu list from Czapiński and Barnes  [7]. Unlike SIMD-TS presented in  [29], the algorithm proposed in this paper is deterministic and it benefits from communication between parallel Tabu Search instances. Section  2 presents development of Parallel Multistart Tabu Search method. GPU implementation of PMTS is described in Section  3, then in Section  4 results of computational experiments are presented and discussed. Finally, conclusions are presented in Section  5.