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The Transformer Network for the Traveling Salesman Problem (arxiv.org)
49 points by djoldman on Mar 20, 2021 | hide | past | pdf | 20 comments on HN

In plain words: A language-model-style network builds a route through every city, learning by trial and reward instead of from solved examples, then searches several candidate routes to pick the best. It beat other AI-trained route planners, finishing 100-city tours just 0.39% worse than the best route.

Abstract

The Traveling Salesman Problem (TSP) is the most popular and most studied combinatorial problem, starting with von Neumann in 1951. It has driven the discovery of several optimization techniques such as cutting planes, branch-and-bound, local search, Lagrangian relaxation, and simulated annealing. The last five years have seen the emergence of promising techniques where (graph) neural networks have been capable to learn new combinatorial algorithms. The main question is whether deep learning can learn better heuristics from data, i.e. replacing human-engineered heuristics? This is appealing because developing algorithms to tackle efficiently NP-hard problems may require years of research, and many industry problems are combinatorial by nature. In this work, we propose to adapt the recent successful Transformer architecture originally developed for natural language processing to the combinatorial TSP. Training is done by reinforcement learning, hence without TSP training solutions, and decoding uses beam search. We report improved performances over recent learned heuristics with an optimal gap of 0.004% for TSP50 and 0.39% for TSP100.

Xavier Bresson, Thomas Laurent
arXiv:2103.03012 · cs.LG · submitted Mar 4, 2021
abstract · pdf · html

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thxg on Mar 20, 2021 [–]
For context, the tests here are performed on 50- and 100-city instances, as they are limited by current GPU memory sizes. Meanwhile, the best heuristic code for TSP [1] frequently provides optimal solutions to 100k-city instances ("heuristic" = no optimality guarantee, like what this paper proposes; but one can then seek a proof of optimality if desired using a slower "exact" algorithm).

Also, Dantzig, Fulkerson and Johnson solved a 50-city TSP to optimality, with an exact method, by hand, in 1954 [2]. The practical running time was slower than what is proposed here, admittedly :-).

This is not a criticism of the paper, though: To their credit, the authors are quite straightforward about this, they're not trying to hide it. Their point is to demonstrate that their machine learning approach has potential.

[1] http://webhotel4.ruc.dk/~keld/research/LKH/

[2] http://www.math.uwaterloo.ca/tsp/uk/history.html

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algo_trader on Mar 20, 2021 | parent [–]
> best heuristic code for TSP .. 100k-city instances

These "best cases" are hand-tuned for the domain by algorithm experts - right? How big is the performance jump from generic z3 or whatever?

Edit: After alphazero, there was lots of chatter about improving combinatorical problems. This has proven elusive.

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bopbeepboop on Mar 21, 2021 | parent [–]
My impression of Alpha Zero research is that given enough feedback, AI would find “good” routes in the sense of sensible driving, minimal lefts, nearby bathrooms, etc — but not that it had anything in particular to say about hard combinatorics.

Ie, AI would find “fast” routes in the sense of “a human driving these gets done fast, all things considered” but not necessarily “fast” in a simple combinatorics problem — which is actually good, because simple combinatorics often fails in the real world (eg, longer routes without left turns are faster).

Did I miss something about how this more directly applies to combinatorics?

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algo_trader on Mar 21, 2021 | parent [–]
Combinatorics (and AZ) can have unlimited self-play (i.e. search) with precise reward. This is quite rare in most real world problems.

I dont really have a strong intuition for mathematics. It is possible that nets are simply good with spatial boards and not good enough with arbitrary graphs.

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thxg on Mar 21, 2021 | parent [–]
> These "best cases" are hand-tuned for the domain by algorithm experts - right?

Correct. It is arguably more than just "hand-tuned". The algorithm I mentioned, LKH, was entirely designed for TSP, and I'm not aware of it being applicable to much else.

> How big is the performance jump from generic z3 or whatever?

For TSP, an enormous jump. Certainly more than 3 orders of magnitude.

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alfiedotwtf on Mar 21, 2021 | parent [–]
Provably optimal? I thought TSP was NP-Hard and could only ever be solved via heuristics
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gopiandcode on Mar 21, 2021 | parent [–]
NP-hard refers to solving the problem in the general case - for a given specific instance it may be possible to find an optimal solution more easily by exploiting specific features of the graph (for instance, a series of nodes arranged in a ring would have a trivial optimal solution).
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djoldman on Mar 20, 2021 [–]
"We report improved performances over recent learned heuristics with an optimal gap of 0.004% for TSP50 and 0.39% for TSP100."

Complexity: O(n^2)

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mpoteat on Mar 20, 2021 | parent [–]
If true, what implications does this have for P = NP? My intuition for a long time has been that P = NP only in the limit of an "infinitely sophisticated" algorithm. Such that as we do more research (or I suppose train larger models), we get closer and closer to the polynomial ideal.
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knuthsat on Mar 20, 2021 | parent [–]
Current best heuristics, Lin-Kernighan-Helsgaun arrive probabilistically at better percentages for TSP.

Definitely makes you wonder if P = NP is even relevant when all the interesting instances can be solved to optimality or very near to it extremely often.

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UncleMeat on Mar 21, 2021 | parent [–]
Some NP-complete problems have provably bad approximations if P!=NP. You don't get the same polynomial reduction behavior if you are using approximation algorithms. So for some problems we may have utterly practical algorithms and for others we never will (assuming P!=NP).
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visarga on Mar 20, 2021 | parent [–]
Doesn't apply here because this algorithm is an approximate TSP.
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greesil on Mar 20, 2021 | parent [–]
According to the website below, the DP solution is O(n^2 * 2^n). That would be a big speedup.

https://www.geeksforgeeks.org/travelling-salesman-problem-se...

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p1necone on Mar 21, 2021 [–]
When I read Transformer I thought this was some sort of electrical circuit based solution. I wonder if you could get some sort of solution by representing the network as an electrical circuit.
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mandelken on Mar 21, 2021 | parent [–]
You could try it on a quantum computer, but you'll have to wait a decade (at least) the see one with enough bits to solve a problem of practical size ;-)
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krastanov on Mar 21, 2021 | parent [–]
We do not have reasons to believe that a quantum computer would be much better than classical for this class of problems.
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ampdepolymerase on Mar 21, 2021 | parent [–]
You can build an analogue simulator for TSP by exploiting the properties of the path integral.
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amelius on Mar 20, 2021 [–]
I'd like to see how it performs on trivial instances like a NxM grid of points (perhaps with small perturbations).
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Louis_Sarwal on Mar 21, 2021 [–]
I believe that AI cannot solve this problem by generating a better algorithm. The only algorithm that will give the most optimal result is brute force, trying every possible combination. Any form of deep learning will only do as well as it gets closer to that, and require just as much computation. The advent of working quantum computers and other hardware gains will be the only solution. There is no software that can solve the traveling salesman problem, nor will there ever be, without much better hardware.
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krastanov on Mar 21, 2021 | parent [–]
I mentioned this in another thread: currently we do not have reasons to believe quantum computers are better than classica at solving NP problems.
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