
One-Way Salesman Finds Fast Path Home - jonbaer
https://www.quantamagazine.org/one-way-salesman-finds-fast-path-home-20171005/
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graycat
There's an old, nice, simple, somewhat similar piece of work by Karp: For the
cities, use one of the two usual, simple, fast algorithms to find a minimum
spanning tree, and then do a depth first traversal of the tree just declining
to revisit a city already visited. Turns out that in Euclidean spaces (right,
the symmetric problem) this simple device has some astoundingly good
convergence properties as the number of cities grows. Having lots of cities is
where the usual combinatorial optimization techniques, worst case, go
exponential, and the work by Karp works great in just those cases.

In effect, roughly, due to the subtrees nearest the leaves, Karp's technique
also works with _clusters_ of cities maybe something like in the OP.

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rocqua
This [1] seems to be the paper they are referring to on arxiv.

[1] [https://arxiv.org/abs/1708.04215](https://arxiv.org/abs/1708.04215)

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nagVenkat
> The computation of the single best answer for what is known as the traveling
> salesman problem is famously infeasible.

I think this statement is not true. TSP can be solved to optimality when the
number of cities is small^. GuRoBi and CPLEX's mixed integer solvers can solve
TSPs when the number of cities is small. In fact [1] gives the largest TSPs
solved. Now solving them fast is a totally different question and I feel like
this distinction needs to be made

^ in a reasonable amount of time.

[1]
[http://www.math.uwaterloo.ca/tsp/pla85900/index.html](http://www.math.uwaterloo.ca/tsp/pla85900/index.html)

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excalibur
85,900 is a small number of cities? The US has 19,354 "incorporated places",
which balloons to around 35,000 if you include unincorporated towns.

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grogenaut
You know tsp doesn't just mean cities or actual sales people right?

It can apply to lots of things like 3d printer head or other toolpath
optimization and hundreds of other problems maybe millions

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bowmessage
Obviously -- but that's not what the article is arguing is impossible.

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freyir
Infeasible, not impossible.

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gerdesj
Why can't the asymmetric version of TSP be re-coded into TSP but with twice
the number of nodes, each of which have a "distance" of zero to their
counterparts?

Ahhh, I think I've just noticed the fatal flaw here - it makes the graph
directed. Perhaps fiddling with the notion of +0 and -0 might fix it ... 8)

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Tarrosion
It can, see eg Bill Cook's "in pursuit of the traveling salesman." I've also
wondered how this approximation algorithm relates to the standard asymmetric
to symmetric transformation. I should read the paper I guess.

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scott00
It appears to me that what they've accomplished is to devise an algorithm that
can, in polynomial time, find a solution to the ATSP with cost no worse than a
constant multiple of the optimal solution.

Can any domain experts confirm or deny my interpretation?

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sanxiyn
This is correct.

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Sniffnoy
Worth noting here is that the approximation factor here is 5500. (I.e., the
algorithm is only guaranteed to return a route whose length is no more than
5500 times that of the optimal route.) Much bigger than 3/2! But apparently
it's the first polynomial-time constant-factor approximation algorithm at all
for it, so...

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amelius
Any applications for this algorithm, besides the one mentioned in the article?

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jon-wood
The situation I've had to apply the travelling salesman problem to, and
probably the one its most applicable in, was delivery logistics. Given a list
of deliveries that need to be made, and the time windows in which they'd been
booked, what is the optimal order in which to make those deliveries?

This problem gets particularly interesting when being applied to deliveries
between 4pm and 10pm in London, because you have to start taking into account
things like rush hour traffic, road closures, and football matches. You also
occasionally have to make tradeoffs like deciding whether it's better to
deliver 4 small orders a little late, or one large order _really_ late - for
those situations we could reschedule those orders and allow the routing system
to generate a new route.

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fred_is_fred
I know this is grossly simplifying but it seems the algorithm works exactly
how a sane delivery company would work.

Imagine a company in Paris that has multiple UK deliveries coming up. First
let's lump together all the deliveries in London. Then let's lump together all
the deliveries in South London, then let's lump together everything in
neighborhood Foo in South London. Then the routes you worry about are your
longest ones, like getting to London or into London in the first place. This
makes the assumption that it's quicker to get from a spot in South London to
neighborhood Foo than it is from South London to say Leeds.

It didn't appear to me from this article that any consideration was given to
large late orders or what you mentioned, but that would up the complexity.

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antoncohen
This reminds me of Google's public transport routing algorithm.

[https://research.googleblog.com/2016/03/an-update-on-fast-
tr...](https://research.googleblog.com/2016/03/an-update-on-fast-transit-
routing-with.html)

