The Gödel Prize, TSP, and volume growth

Recently, Sanjeev Arora and Joe Mitchell won the Gödel prize for their work on Euclidean TSP. They show that given n points in \mathbb R^2, and a parameter \varepsilon > 0, it is possible to compute, in polynomial time, a traveling salesman tour of the input whose length is at most a factor 1+\varepsilon more than the length of the optimum tour. (This is called a polynomial-time approximation scheme, or PTAS.)

Later, Arora extended this to work in \mathbb R^k for every fixed k \geq 2. What properties of \mathbb R^k are really needed to get such an algorithm?

Certainly a key property is that the volume of a ball of radius r in \mathbb R^k only grows like O(r^k). This ensures that one can choose an \epsilon-net of size at most O(1/\epsilon)^k in a ball of radius {O(1)}, which is essential for using dynamic programming. In my opinion, this leads to the most fascinating problem left open in this area:

Is bounded volume growth the only property needed to get a PTAS?

This would imply that the use of Euclidean geometry in Arora’s algorithm is non-essential. We can state the question formally as follows. Let (X,d) be a metric space, and let \lambda(X,d) be the smallest number \lambda such that for every r > 0,every ball of radius r in X can be covered by \lambda balls of radius r/2. Is there a PTAS for TSP in X? (In other words, the running time should be bounded by a polynomial in n =|X| whose degree depends only on \lambda(X,d).)

The problem is open, though there are some partial results by Talwar.

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