Extremely Fast “Solution” to the Large-Scale and Very Large-Scale Vehicle Routing Problem

Abstract

A solution to the vehicle routing problem (VRP) is presented that takes only quadratic space, O(n2), and quadratic time, O(n2), if n is the number of stops on a route. The input is assumed to be a list of stops of length n in longitude, latitude format. The output is an origin-destination (OD) matrix of size O(n2), which takes O(n2) time to build. The element (i, j) in the matrix is the approximate driving distance between stop i and stop j on the route. Each approximate driving distance takes constant or O(1) time to compute. (The approximate driving distance appears in previous work by the author, published in URISA GIS-Pro ‘19 and CalGIS 2020.) This OD matrix is well-suited for solving large-scale and very large-scale VRP problems, since computing approximate driving distances is lightning fast. For instance, using real-world data, it took less than one (1) second to produce a route with 5,156 stops. The OD matrix can be used with any exact or approximation algorithm to find a route, including the nearest-neighbor approximation algorithm: Starting at an origin, the next closest stop is visited repeatedly, ending at the destination once all stops have been visited. Determining the next stop to visit takes linear or O(n) time to compute, and this is done O(n) times. This solution to the VRP is a polynomial-time, O(n2), approximation; it is not exact, but is extremely fast.