One of the most intriguing areas of operations research is routing, in which the goal is to find efficient paths for transporting items through a complex network. The network is usually represented by a graph like the one below.

Each node represents a location, and a route is a path through a set of nodes.

Most real routing problems involve finding efficient routes for vehicles — cars, trains, airplanes, and so on — so they are often referred to as vehicle routing problems.

Routing problems can be divided into two main types — node-routing problems and arc-routing problems — depending on whether the goal is to visit the nodes (locations) or the arcs (the edges connecting them). We'll give an example of each.

First, here's an arc-routing problem that Google needs to solve daily. If you have seen the Google Maps Street View, you may have wondered how Google obtains street-level images at millions of addresses around the world. The answer is simple: a Google team continually drives the world's roads in a fleet of vehicles equipped with cameras that automatically take pictures of each address. Google's problem is to construct the shortest route for each vehicle that traverses every street in an assigned region. If you represent this problem by a graph, in which arcs represent streets, and nodes are street intersections, the arc-routing problem is to find the shortest path that traverses every arc in the graph. Google uses the same technology provided in OR-Tools to solve this problem every day.

An example of a node-routing problem is vehicle routing. Suppose that a company needs to deliver packages to various locations, using a fleet of vehicles. In the graph for this problem, nodes represent locations, and arcs represent routes between them. Each arc has a weight, corresponding to the cost of traveling that route. The problem: find a set of paths in the graph (corresponding to delivery routes for each vehicle) that includes every destination while minimizing the total cost. This differs from the arc-routing problem because the paths don't have to traverse every arc, just include every node.

OR-Tools includes a specialized routing library to solve many types of vehicle routing problems:

You can also take a look at other common routing tasks and options for the routing solver.

Limitations on solving node-routing problems

Node-routing problems, such as TSPs and VRPs, are inherently intractable: the length of time it takes to solve them grows exponentially with the size of the problem. For sufficiently large problems, it could take OR-tools (or any other routing software) years to find the optimal solution. As a result, OR-Tools sometimes returns solutions that are good, but not optimal. To find a better solution, change the search options for the solver. See Changing the search strategy for an example.

Note: We should add that there are other solvers, such as Concorde, dedicated to solving very large TSPs to optimality, which surpass OR-Tools in that area. However, OR-Tools provides a better platform for solving more general routing problems that contain constraints beyond those of a pure TSP.