A Conjecture on random bipartite matching

In this note we put forward a conjecture on the average optimal length for bipartite matching with a finite number of elements where the different lengths are independent one from the others and have an exponential distribution. The problem of random bipartite matching (or assignment) is interesting both from the point of view of optimisation theory and of statistical mechanics [1, 2, 3, 4]. Each instance of the problem is characterised by a N cross N matrix d; sometimes d(i, k) represents the distance between i and k. We are interested to compute the length of the optimal bipartite matching defined as: L(d) = min Π ∑ i=1,N d(i,Π(i)), (1) where the minimum is done over all the N ! permutations (Π) of N elements and Π(i) denote the results of the action of the permutation Π on i. If the matrix d has a given probability distribution, we can define the average of L(d) over this probability distribution. The problem of computing this average can be studied using the technique of statistical mechanics. A well studied case is when the elements of d are statistically independent one from the others. Here we can use replica theory [5, 6] in the mean field approximation. When the probability distribution of each element of d is flat in the interval [0, 1], after a long computation it was found [2, 3] that for large N we have: 〈L〉N = ζ(2)− ζ(2) + 2ζ(3) N +O ( 1 N )