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Message: [QUOTE="JSH, post: 3597613"] That works for me! The idea is to be as complete as possible, as I assume that there would be cases where paths came out with equal values. That's a more general assumption than what she made and sounds good intuitively but you haven't even begun to give any kind of proof. But it is a route to a counterexample to this idea: use two "traps" and prove that the algorithm as described cannot give an optimal path. Goes back to when I was arguing that wrapped up in the real world with any TSP problem is some notion of distance. For instance, long trips tend to cost more, so it costs more to fly to a distant city than to drive to the supermarket. But, as was noted you can have situations where cost has no connection with distance, and you can have a TSP problem where no distances are given, which lead me to the distance normalized algorithm where all nodes are assumed to be equidistant from each other. However, the algorithm I use at its heart is about distance correlation, where the longer the distance between nodes the greater the cost. So if I'm right, it easily solves TSP when cost matches well with distance, and has problems if it does not, like with the "traps" you mention. I call a graph where distance matches perfectly with cost, a perfect graph. That would be an interesting counterexample. I have suspicions for why your intuition is wrong in this case, but of course, I want you to be wrong, so that doesn't say much. But my thinking has to do with rotations in space (don't ask the dimension), so that there is always an angle--if an optimal path exists--that cuts through optimally, or maybe I'm just babbling. Trouble with from scratch problem solving is it can take time, which is why I sound differently this week than last, as last week, I'd just come up with this approach. Kind of just came fully formed--why not use two travelers with one going forward and one going backwards in time? So this entire algorithm is like a baby, and everything is fresh ground as I slowly work my way through theories, implications, rough guesses and complete b.s. in the hopes of getting an answer. James Harris [/QUOTE]

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