Automated drawing of metro maps

Abstract

This work investigates the problem of drawing metro maps which is defined as follows. Given a planar graph G of maximum degree 8 with its embedding and vertex locations (e.g. the physical location of the tracks and stations of a metro system) and a set L of paths or cycles in G (e.g. metro lines) such that each edge of G belongs to at least one element of L, draw G and L nicely. We first specify the niceness of a drawing by listing a number of hard and soft constraints. Then we show that it is NP-complete to decide whether a drawing of G satisfying all hard constraints exists. In spite of the hardness of the problem we present a mixed-integer linear program (MIP) which always finds a drawing that fulfills all hard constraints (if such a drawing exists) and optimizes a weighted sum of costs corresponding to the soft constraints. We also describe some heuristics that speed up the MIP and we show how to include vertex labels in the drawing. We have implemented the MIP, the heuristics and the vertex labeling. For six real-world examples we compare our results to official metro maps drawn by graphic designers and to the results of previous algorithms for drawing metro maps.