Selberg integrals in 1D random Euclidean optimization problems

We consider a set of Euclidean optimization problems in one dimension, where the cost function associated to the couple of points x and y is the Euclidean distance between them to an arbitrary power p >> 1, the points are chosen at random with uniform measure. We derive the exact average cost for the random assignment problem, for any number of points using Selberg’s integrals. Some variants of these integrals enable the exact average cost for the bipartite travelling salesman problem to be derived.