A reactive GRASP with path relinking for capacitated clustering

摘要

This paper presents a greedy randomized adaptive search procedure (GRASP) coupled with path relinking (PR) to solve the problem of clustering?n nodes in a graph into?p clusters. The objective is to maximize the sum of the edge weights within each cluster such that the sum of the corresponding node weights does not exceed a fixed capacity. In phase?I, both a heaviest weight edge (HWE) algorithm and a constrained minimum cut algorithm are used to select seeds for initializing the?p clusters. Feasible solutions are obtained with the help of a self-adjusting restricted candidate list that sequentially guides the assignment of the remaining nodes. At each major GRASP iteration, the list length is randomly set based on a probability density function that is updated dynamically to reflect the solution quality realized in past iterations. In phase?II, three neighborhoods, each defined by common edge and node swaps, are explored to attain local optimality. The following exploration strategies are investigated: cyclic neighborhood search, variable neighborhood descent, and randomized variable neighborhood descent (RVND). The best solutions found are stored in an elite pool. In a post-processing step, PR is applied to the pool members to cyclically generate paths between each pair. As new solutions are uncovered, a systematic attempt is made to improve a subset of them with local search. Should a better solution be found, it is saved temporally and placed in the pool after all the pairs are investigated and the bottom member is removed. The procedure ends when no further improvement is possible. Extensive computational testing was done to evaluate the various combinations of construction and local search strategies. For instances with up to 40 nodes and 5 clusters, the reactive GRASP with PR found optimal solutions within a negligible amount of time compared to CPLEX. In general, the HWE algorithm in the construction phase, RVND in the local search phase, and the use of PR provided the best results. The largest instances solved involved 82 nodes and 8 clusters.