ON THE STOCHASTIC EUCLIDEAN TRAVELING SALESPERSON PROBLEM FOR DISTRIBUTIONS WITH UNBOUNDED SUPPORT

We study the asymptotic behavior of the shortest tour T(n) through n points X1,...,X(n) in R(d) (d greater-than-or-less-than 2), where (X(i))i greater-than-or-equal-to 1 are i.i.d. random variables, whose density f(x) has an unbounded support. Beardwood et al. [2] conjectured that T(n)/n1-1/d converges a.s. if and only if integral f(x)1-1/d dx < infinity and integral f(x)\\x\\d/(d-1) dx < infinity. We disprove this conjecture, and we show that the second integrability condition is not strong enough. We give a stronger condition that is optimal (but not necessary).