An Accelerated-Limit-Crossing-Based Multilevel Algorithm for the $p$-Median Problem

In this paper, we investigate how to design an efficient heuristic algorithm under the guideline of the backbone and the fat, in the context of the p-median problem. Given a problem instance, the backbone variables are defined as the variables shared by all optimal solutions, and the fat variables are defined as the variables that are absent from every optimal solution. Identification of the backbone (fat) variables is essential for the heuristic algorithms exploiting such structures. Since the existing exact identification method, i.e., limit crossing (LC), is time consuming and sensitive to the upper bounds, it is hard to incorporate LC into heuristic algorithm design. In this paper, we develop the accelerated-LC (ALC)-based multilevel algorithm (ALCMA). In contrast to LC which repeatedly runs the time-consuming Lagrangian relaxation (LR) procedure, ALC is introduced in ALCMA such that LR is performed only once, and every backbone (fat) variable can be determined in O(1) time. Meanwhile, the upper bound sensitivity is eliminated by a dynamic pseudo upper bound mechanism. By combining ALC with the pseudo upper bound, ALCMA can efficiently find high-quality solutions within a series of reduced search spaces. Extensive empirical results demonstrate that ALCMA outperforms existing heuristic algorithms in terms of the average solution quality.