On the Restricted Steiner Multi Cycle Problem

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We introduce a new problem, called Steiner Multi Cycle Problem, in which we are given a complete weighted graph G=(V,E), which respects the triangle inequality, a collection of terminal sets {T1,…,Tk}, where for each a in [k] we have a subset Ta of V and these terminal sets are pairwise disjoint. The goal is to find a set of disjoint cycles of minimum cost such that for each a in [k], all vertices of Ta belong to a same cycle. Our main interest is in a restricted case where |Ta|=2, for each a in [k], which models a collaborative problem with pickup and delivery. We show that even the restricted problem is NP-Hard, and present three heuristics to solve it: a dynamic programming algorithm called Refinement Search Heuristic, which explores geometric and laminar set properties; a heuristic based on the Gonzales Clustering algorithm; and a GRASP Heuristic with Path Relinking. We performed computational experiments with 525 instances to compare these methods, which achieved results close to the optimum, highlighting the performance of the Refinement Search Heuristic both in quality and time.