Abstract
The authors study the problem of a travelling salesman who must visit a randomly chosen subset of sites of a d-dimensional lattice. The average length of the shortest path per chosen site is alpha (q) where (1-q) is the density of chosen sites. For a triangular lattice, they show that alpha (q) differs from 1 only by terms of order q5. For the square lattice, they show that, to first order in q, optimal paths can be found from the dynamics of a model of a one-dimensional gas of kinks and antikinks. The authors find alpha (q)<or=1+terms of order q3/2. They also obtain a constructive upper bound valid for all q, which gives alpha (q)>or=L/3(1-q)-1/2 as q tends to 1.