"Tristan Miller" wrote in message
...
Greetings.

What about the easier problem of the shortest journey where every *line*
is
visited? That is, what is the shortest journey such that the traveller
uses some portion of each of the Bakerloo, Central, Circle, District, East
London, Hammersmith & City, Jubilee, Metropolitan, Northern, Picadilly,
Victoria, and Waterloo & City lines at least once?

For example, here's one journey which attempts to satisfy definitions
(c.3), though I don't know if it's really the shortest:

1. King's Cross St. Pancras (Victoria, Northern, Hammersmith, Circle,
Metropolitan, Picadilly) to Whitechapel (East London, District) on the
Hammersmith & City Line: 6 stops, including Liverpool Street (Central)

2. Whitechapel to Liverpool Street on the Hammersmith & City Line: 2 stops

3. Liverpool Street to Bank (Waterloo & City) on the Central Line: 1 stop

4. Bank to Waterloo (Jubilee, Bakerloo) on the Waterloo & City Line: 1
stop

But you aren't really going through each line with that. You're going
through stations that have the required lines. Not the same thing surely.