Comments on: The Traveling Salesman Problem: An Optimization Model

By: gold price

gold price — Thu, 02 Aug 2012 17:57:15 +0000

The actual problem is, on its surface, a simple one. However, current methods require each instance to be solved individually. A general solution has eluded mathematicians for decades. This difficulty is due to the fact that the TSP is an NP-hard problem. This means that, among other things, the TSP is a problem in which correct solutions are easy to verify, but there is no efficient way to solve the problem itself. This also means that a solution to the TSP would put to rest the question of whether or not it is possible to solve NP-complete problems. Called the P versus NP problem, this is one of the most challenging questions facing mathematicians today. In fact, it was designated a millennium prize problem by the Clay Mathematics Institute. This means that the one who solves this problem will win a million dollar prize.

By: Gordon Royle

Gordon Royle — Sun, 22 Jul 2012 02:43:03 +0000

Just picking a nit… the NP tag on its own does not mean that the problem is necessarily hard; all polynomial time or even trivial decision problems are in NP.

It is the NP-hard or NP-complete problems that cause difficulties.

By: Bill Fahle

Bill Fahle — Fri, 20 Jul 2012 19:26:35 +0000

An interesting result I saw recently had to do with the so-called gas station problem. It’s a case where adding a realistic constraint to the problem makes it solvable.
http://www.cs.umd.edu/projects/gas/gas-station.pdf

By: Laurent

Laurent — Fri, 20 Jul 2012 15:06:59 +0000

I didn’t mean mathematicians around the world should stop looking for a general solution, Peter.
All I meant is that, in my very specific, pointy-haired environment, if someone puts the NP tag on an approach, it’s the kiss of death. We won’t consider any approximation, any heuristics…anything, we will move right along and try to sidestep the “problem”, as if it were the plague.
What I meant is that, even though some problems are truly NP, under certain conditions (where n is kept below a limit), they are tractable and can be addressed quite elegantly using an approximation or a heuristics.
In my environment, which is not focused on mathematics, no one should read the NP sign as meaning “dead end”.

In that spirit, could you share a link to that talk of last December?

By: Peter Cameron

Peter Cameron — Fri, 20 Jul 2012 06:31:28 +0000

I don’t agree. I think that the world is big enough for the Good and the Best to coexist.

I have been to several conference talks (most recently just last December) where somebody presented a new and improved heuristic for TSP. Even if there were an abstract proof that P=NP, this activity would continue.

A similar case fairly recently was the proof that primality testing can be done in polynomial time. It was certainly not the case that people immediately stop using probabilistic algorithms for primality testing. Or again, Khachian’s ellipsoid method showed that linear programming is polynomial-time, but a lot of people went on using the simplex method (and still do).

Another point is that complexity theory is wide enough to embrace approximation. There are some cases where, even though exact calculation is hard, there is an efficient approximation algorithm; but many more where this is not known.

By: Laurent

Laurent — Thu, 19 Jul 2012 14:36:12 +0000

Sometimes, Best is the enemy of Good (Enough). Sure, a general solution would be marvelous, but a good computational heuristics for cases where n is < 1000 would be tremendously useful to a lot of applications.
As it stands, the NP scarecrow turns everyone away on the basis that no general solution has been found…leaving truck drivers in the neighborhood to a random walk of fuel-inefficient deliveries.