Category: Complexity

Watching whales well – the travelling salesman problem

An aerial photograph of São Miguel lighthouse in the Azores showing the surrounding tree-covered cliff and winding road.

by Paul Curzon, Queen Mary University of London

Photograph of a whale's tail emerging from the deep blue sea with a tree-covered cliff behind it.
Whale tail photograph by Kevin O’Leary from Pixabay

Sasha owns a new tour company and her first tours are to the Azores, a group of volcanic islands in the Atlantic Ocean, off the coast of Portugal. They are one of the best places in the world to see whales and dolphins, so lots of people are signing up to go.

Sasha’s tour as advertised is to visit all nine islands in the Azores: São Miguel, Terceira, Faial, Pico, São Jorge, Santa Maria, Graciosa, Flores and Corvo. The holidaymakers go whale watching as well as visiting the attractions on each island, like swimming in the lava pools. Sasha’s first problem, though, is to sort out the itinerary. She has to work out the best order to visit the islands so her customers spend as little time as possible travelling, leaving more for watching whales and visiting volcanos. She also doesn’t want the tour to go back to the same island twice – and she needs it to end up back at the starting island, São Miguel, for the return flight back home.

Trouble in paradise

It sounds like it should be easy, but it’s actually an example of a computer science problem that dates back at least to the 1800s. It’s known as ‘The Travelling Salesman Problem’ because it is the same problem a salesman has who wants to visit a series of cities and get back to base at the end of the trip. It is surprisingly difficult.

It’s not that hard to come up with any old answer (just join the dots!), but it’s much tougher to come up with the best answer. Of course a computer scientist doesn’t want to just solve one-off problems like Sasha’s but to come up with a way of solving any variant of the problem. Sasha, of course, agrees – once she’s sorted out the Azores itinerary, she then needs to solve similar problems, like the day trip round São Miguel. Her customers will visit the lakes, the tea factory, the hot spring-fed swimming pool in the botanic gardens and so on. Not only that, once Sasha’s done with the Azores, she then needs to plan a wildlife tour of Florida. Knowing a quick way to do it would help her a lot.

The long way round

An aerial photograph of São Miguel lighthouse in the Azores showing the surrounding tree-covered cliff and winding road.
Photo of São Miguel lighthouse Image by Vianney Dugrain from Pixabay

No one has yet come up with a good way to solve the Travelling Salesman problem though and it is generally believed to be impossible. You can find the best solution in theory of course: just try all the alternatives. Sasha could first work out how long it is if you go São Miguel, Terceira, Faial, Pico, São Jorge, Santa Maria, Graciosa, Flores, Corvo and back to São Miguel, then work out the time for a different order, swapping Corvo and Flores, say. Then she could try a different route, and keep on till she knew the length of every variation. She would then just pick the best. Trouble is, that takes forever.

Even this small problem with only 9 islands has over 20 000 solutions to check. Go up to a tour of 15 destinations and you have 43 billion calculations to do. Add a few more and it would take centuries for a fast computer running flat out to solve it. Bigger still and you find the computer would have to run for longer than the time left before the end of the universe. Hmmm. It’s a problem then.

Be greedy

The solution is not to be such a perfectionist and accept that a good solution will have to be good enough even though it may not be the absolute best. One way to get a good solution is called using a ‘greedy’ algorithm. You start at São Miguel and just go from there to the nearest island, from there to the nearest island not yet visited, and so on till you have done them all. That would probably work well for the Azores as they are in groups, so visiting the close ones in each group together makes sense. It doesn’t guarantee the best answer in all cases though.

Or just go climb a hill

Another way is to use a version of ‘hill climbing’. Here you take any old route and then try and optimise it, by just making small changes – swapping pairs of legs over, say: instead of going Faial to Pico and later Corvo to Flores try substituting Pico to Flores and Faial to Corvo, with the rest the same but in the opposite order. If the change is an improvement keep it and make later changes to that. Otherwise stick with the original. Either way keep trying changes on the best solution you’ve found so far, until you run out of time.

So Sasha may want to run a great tour company but there may not be enough time in the universe for her tours to be guaranteed perfect…unless of course she keeps them very small. After all, just visiting São Miguel and Terceira makes a great holiday anyway.

This article was originally published on the CS4FN website and a copy can also be found on pages 14-15 of issue 10 of the CS4FN magazine, which you can downloads as a PDF below. All of our free material can be downloaded here.

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This blog is funded through EPSRC grant EP/W033615/1.

Tantrix: P=NP?

You can find computer science everywhere – even in popular Solitaire games and puzzles. Most people have played games like Tetris, Battleships, Mastermind, Tantrix and Minesweeper at some point. In fact, all these games have a link to one of the deepest, fundamental problems remaining in Computer Science. They are all linked to a famous equation that is to do with the ultimate limitations of computers.

A 5 tile Tantrix puzzle to solve

The sciences have many iconic equations that represent something fundamental about the world. The most famous is of course Einstein’s E=mc², which even non-scientists have heard of. The most famous equation in computer science is ‘P=NP’. The only trouble is no one has yet proved whether it is true or not! There is even a million dollar prize up for grabs for anyone who does, not to mention great fame!

P=NP boils down to the difference between checking if someone’s answer to a puzzle is correct, as against having to come up with the answer in the first place.

You are so NP!

Computer Scientists call problems where it is easy to check answers ‘NP problems’. Ones where it’s also easy to come up with solutions are ‘P problems’. So P=NP, if it were true, would just mean that all problems that are easy to check are also easy to solve.

Let’s take Tantrix rotation puzzles to see what it is all about. Tantrix is a popular domino-type game using coloured hexagonal pieces like the ones in the image. The idea of a Tantrix rotation puzzle is that you place some tiles randomly on the table in a connected pattern. You are then not allowed to move the position of any piece. All you can do is rotate them on the spot. The problem is to rotate the pieces so that all the coloured lines match where tiles meet – red to red lines, blue to blue lines and so on.

Have a go at the Tantrix rotation puzzle above before you read on.

Easy to check?

A solution is here if you want to check it. In fact you can quickly check any claimed rotation puzzle ‘solution’. All you do (the checking ‘algorithm’) is look at each tile in turn and check each of its edges does match the edge of the tile it touches, if any. If you find a tile edge that doesn’t match then the ‘solution’ isn’t a solution after all. How long would that take to check? With say a 10 piece puzzle there are 10 pieces to check each with 6 edges so in total that is 10×6 = 60 things to check. That wouldn’t take too long. Even with 100 pieces it would be only 600 things to check – 10 minutes if you could check an edge a second. So Tantrix rotation puzzles are NP puzzles – they can be checked quickly.

But can you solve it?

The question is: “Are rotation puzzles P puzzles too?” Can they always be solved quickly if you or a computer were clever enough? You may have found that puzzle easy to solve. It is much harder to come up with a quick way that is guaranteed to solve any rotation puzzle I give you. One way would be to methodically work through every combination of tile rotations to see if it worked. That would take a long time though.

There are 6 positions for the first piece (ways to rotate it), but for each of those 6 positions the second piece could be in 6 positions too … and so on for each other piece. Altogether for a 10-tile puzzle there are 6x6x6x6x6x6x6x6x6x6 (i.e., over 60 million) positions to check looking for a solution (and we might have to check them all). If you could check one position a second, it would take you around 700 days non-stop (no eating or sleeping). That is just for a 10-tile puzzle…now for a 100-tile puzzle – I’ll leave you to work out how long that would take.

It is not what computer scientists call “quick”.

How clever do you have to be?

If P=NP is true it would mean there is a quick way of solving all Tantrix rotation puzzles out there, if only someone were clever enough to think of it. If P=NP is not true then it might just not be possible however clever you are. Trouble is no-one knows if it’s true or not…

Tantrix: Solve one, solve them all

We’ve seen how Tantrix rotation puzzles can show what we mean by the question “Is P=NP?” It turns out Tantrix rotation puzzles are also something called ‘NP-complete’. That means it is one of a special kind of problem.

NP-complete problems are the really hard ones. Some are also really important to be able to solve quickly. For example, suppose you are a taxi driver and wanted your SatNav to be able to quickly work out the fastest circular route that went via a whole series of places you wanted to visit (in any order), getting you back home. Simple as it sounds, it is an NP-complete problem. It can’t be done quickly even by a fast computer. The best that can be done is to come up with a good answer not a perfect one – using what’s known as a ‘heuristic’. There are lots of ways of coming up with such good answers – using Swarm Intelligence is one way, for example. The point is none can guarantee the best answer every time.

NP-completeness is important because if you could come up with a quick algorithm for solving one NP-complete problem, computer scientists know how to convert such an algorithm into one that solves all the other NP problems…P would equal NP…Trouble is no one has so far done it…

– Paul Curzon, Queen Mary University of London, 2007

Play Tantrix online at the official site.