Adaptive memory programming for matrix bandwidth minimization

Abstract

In this paper we explore the influence of adaptive memory in the performance of heuristic methods when solving a hard combinatorial optimization problem. Specifically, we tackle the adaptation of tabu search and scatter search to the bandwidth minimization problem. It consists of finding a permutation of the rows and columns of a given matrix which keeps the non-zero elements in a band that is as close as possible to the main diagonal. This is a classic problem, introduced in the late sixties, that also has a well-known formulation in terms of graphs. Different exact and heuristic approaches have been proposed for the bandwidth problem. Our contribution consists of two new algorithms, one based on the tabu search methodology and the other based on the scatter search framework. We also present a hybrid method combining both for improved outcomes. Extensive computational testing shows the influence of the different elements in heuristic search, such as neighbourhood definition, local search, combination methods and the use of memory. We compare our proposals with the most recent and advanced methods for this problem, concluding that our new methods can compete with them in speed and running time.

Variable neighbourhood search for bandwidth reduction

“…We compare the solution qualities (minimum bandwidth obtained) and running times of our VNS based heuristic with six other approaches from the literature: TS (Tabu Search) [19], GRASP-PR (Greedy Randomized Adaptive Search Procedure-Path Relinking) [24], SS-TS (Scatter Search-Tabu Search) [2], GA (Genetic Algorithm) [16], NS a , NS b (Node Shift) [16], and SA (Simulated Annealing) [25]. Based on previous results reported, the SA heuristic may be seen as the state-of-the-art heuristic.…”

“…Test instances. Following [2,16,17,19,24,25] we compiled 113 instances from the Harwell-Boeing Sparse Matrix Collection (http://math.-nist.gov/MatrixMarket/data/Harwell-Boeing). They are divided into two sets; the first set consists of 33 instances with the number of nodes ranging from 30 to 199 and the second set has 80 instances with the number of nodes ranging from 200 to 1000.…”

“…In Table 5, we summarize the results of all methods on all 113 test instances (made up of 33 small instances and 80 large instances). Results for TS, GRASP-PR, SS-TS, GA, NS and SA are taken from [2] and for PSO-HC from [17]. Detailed results may be found in Appendix 2.…”

“…Campos et al [2] propose two methods. One is based on Tabu Search (TS) and the other on Scatter Search (SS).…”

Metaheuristic-based Heuristics for Symmetric-matrix Bandwidth Reduction: A Systematic Review

“…Campos, Piñana and Martí [5,6] SS-TS 2011 TS-CPM Pop and Matei [45] GP-band 2011 Koohestani and Poli [26] GPHH-band 2011 Mamaghani and Meybodi [37] LA-band 2011 Pintea, Crişa and Shira [43] ACO-PCS2 2012 Isazadeh, Izadkhah and Mokarram [20] LAGA 2012 Kaveh and Sharafi [24] CSS-band 2012 Arathi, Doss and Kanakadurga [2] Arathi-Doss-Kanakadurga 2012…”