The shortest way to visit all metro lines in a city

TL;DR

This paper explores the minimal steps needed to visit all metro lines in Paris and Tokyo using mathematical programming, revealing surprisingly short routes that cover all lines efficiently.

Contribution

It introduces an optimal approach to solve the minimal line-visit problem in metro networks using mathematical programming, with surprisingly concise solutions.

Findings

All 16 Paris metro lines can be visited in 26 steps.

Adding RER lines does not increase the minimal steps.

All 13 Tokyo lines can be visited in 15 steps.

Abstract

What if $\{$a tourist, a train addict, Dr. Sheldon Cooper, somebody who likes to waste time$\}$ wants to visit all metro lines or carriages in a given network in a minimum number of steps? We study this problem with an application to the metro network of Paris and Tokyo, proposing optimal solutions thanks to mathematical programming tools. Quite surprisingly, it appears that you can visit all 16 Parisian metro lines in only 26 steps (we denote by a step the act of taking the metro from one station to an adjacent one). Perhaps even more surprisingly, adding the 5 RER lines to these 16 lines does not increase the size of the best solution. It is also possible to visit the 13 lines of (the dense network of) Tokyo with only 15 steps.