Traveling salesman problem: why visit each city only once?

Solution 1:

Let's look at an example. Suppose you have four cities, A, X, Y, and Z, and A is at distance $1$ from each of the others, while the distance between any two of X, Y, Z is $100$. Then of course a solution to TSP starting at A is given by A to X to A to Y to A to Z to A – let's write this as AXAYAZA, which has length $6$, and visits A multiple times.

Now let's carry out the construction in Rahul's comment. We form a new problem with the same cities but with the distance between cities replaced by the length of the shortest path between the two cities. So in the new problem, A is still at distance $1$ from each of the others, but the distance between any two of X, Y, Z is $2$. Now the tirangle inequality is satisfied (although the configuration is still unrealizable in any Euclidean space), and a solution is given by AXYZA, length $6$, no city visited twice (except for both starting and ending at A).

So we have taken a problem where the solution has multiple visits to a city, and replaced it with an equivalent problem which doesn't. I hope it's clear that you can carry out this construction for any (finite) set of cities and distances. So you can always replace a problem with an equivalent problem where the triangle inequality is satisfied and no city gets visited more than once. So, there is no loss of generality in assuming the triangle inequality holds and no city gets visited more than once. That's why we can restrict our attention to what looks like, but really isn't, a special case.

Solution 2:

Suppose there are roads going directly from every city to every other city. That makes the problem simple enough and general enough to work out some theory around it.
The road AC is shorter than AB+BC, (think of the triangle ABC), unless they are in a straight line. So, if you already visited B earlier, visiting B again is a detour that lengthens the total path.

Solution 3:

I had the same question because I was considering this as a graph problem where distance between two nodes can be any arbitrary number. But I think TSP really means a geometric, physical problem. That is why if you pick two random nodes, the maximum distance between them is limited by the triangle inequality. But even then this is assuming a perfect plane. What if I have this city surrounded by mountains and you cannot go to any other city directly unless you spend days climbing up and down the mountain, but you have one passage that reaches to one other city and you are connected to the rest of the graph thereon? Since you have to stop at where you have started, you would have to visit one city twice to get the best solution.

In other words, if I haven't misunderstood anything, each city would naturally be visited once if map is planar and distances are determined geometrically. But if distances are not bounded by such rules, anything can happen in the optimal solution in my opinion.