The Dyck bound in the concave 1-dimensional random assignment model

Abstract

We consider models of assignment for random N blue points and N red points on an interval of length 2N , in which the cost for connecting a blue point in x to a red point in y is the concave function |x − y| p , for 0 < p < 1. Contrarily to the convex case p > 1, where the optimal matching is trivially determined, here the optimization is non-trivial.The purpose of this paper is to introduce a special configuration, that we call the Dyck matching, and to study its statistical properties. We compute exactly the average cost, in the asymptotic limit of large N , together with the first subleading correction. The scaling is remarkable: it is of order N for p < 1 2 , order N ln N for p = 1 2 , and N arXiv:1904.10867v2 [cond-mat.dis-nn]In this paper we study the statistical properties of the Euclidean random assignment problem, in the case in which the points are confined to a one-dimensional interval, and the cost is a concave increasing function of their distance. The assignment problem is a combinatorial optimization problem, a special case of the matching problem when the underlying graph is bipartite. As for any combinatorial optimization problem, each realisation of the problem is described by an instance J, and the goal is to find, within some space of configurations, the particular one that minimises the given cost function. In the assignment problem, the instance J is a real-positive N × N matrix, encoding the costs of each possible pairing among N blue and N red points (J ij is the cost of pairing the i-th blue point with the j-th red point), the space of configurations is the set of permutations of N objects, π ∈ S N (describing a complete assignment of blue and red points), and the cost function isA random assignment problem is the datum of a probability measure µ N (J) on the possible instances of the problem. The interest is in the determination of the statistical properties of this problem w.r.t. the measure under analysis, and in particular the statistical properties of the optimal configuration. The problem can be formulated as the zero-temperature limit of the statistical mechanics properties of a disordered system, where the disorder is the instance J, the dynamical variables are encoded by π, and the Hamiltonian is the cost function H J (π).The case of random assignment problem in which the entries J ij are random i.i.d. variables, presented already in [1], has been solved at first, in a seminal paper by Parisi and Mézard [2], through the replica trick and afterwards by the Cavity Equations [3] (see also [4] for a recent generalization of those results also at finite system size). The Parisi-Mézard solution also leads to the striking prediction that, calling π opt (J) the optimal matching for the instance J and H opt (J) = H J (π opt (J)) its optimal cost, the average over all instances of H opt , for N large, tends to π 2 6 . This problem is simpler than a spin glass, as the determination of the optimal configuration is feasible in polynomial time (for example, through the celebrated Hungarian Alg...

The Number of Optimal Matchings for Euclidean Assignment on the Line

“…The fact that we can study these two ensembles in parallel is present also in our study for the distribution of the energy distribution of the "Dyck matching", that we perform elsewhere [5,10].…”

“…We are now interested in enumerating the optimal matchings at p = 1 for a fixed configuration J of size N . First of all, we give an alternative representation of J (already adopted in [5]) that will be useful in the following. A configuration J can be encoded by sorting the 2N points in order of increasing coordinate (as in Definition 1 above), and defining -a vector of spacings s(J )…”

“…to √ n! ), and a comparatively smaller number of results has been found so far for the random version of the problem [3,5,10,15]. Note that, remarkably, it is expected that, within this interval, there are two transitions: one at p = 1, with γ (1, 1) = 0, somewhat corresponding to the ordered / non-crossing structural transition, and one at p = 1 2 , with γ ( 1 2 , 1) = 1, corresponding to a transition in which, in the cost of the optimal matching, the leading contribution comes from the few longest edges (of length O(1), when p > 1 2 ) or from the many shortest edges (of length O(N −1 ), when p < 1 2 ), and the logarithmic correction at p = 1 2 comes from the fact that, in this case, all length scales contribute to the leading part of the optimal cost [5].…”

“…), and a comparatively smaller number of results has been found so far for the random version of the problem [3,5,10,15]. Note that, remarkably, it is expected that, within this interval, there are two transitions: one at p = 1, with γ (1, 1) = 0, somewhat corresponding to the ordered / non-crossing structural transition, and one at p = 1 2 , with γ ( 1 2 , 1) = 1, corresponding to a transition in which, in the cost of the optimal matching, the leading contribution comes from the few longest edges (of length O(1), when p > 1 2 ) or from the many shortest edges (of length O(N −1 ), when p < 1 2 ), and the logarithmic correction at p = 1 2 comes from the fact that, in this case, all length scales contribute to the leading part of the optimal cost [5]. -For p < 0, due to the fact that an overall positive factor in c(x) is irrelevant in the determination of the optimal matching, it is questionable if one should consider the analytic continuation of the cost function c(x) = x p (which has the counter-intuitive property that the preferred links are the longest ones), or of the cost function c(x) = p x p (which has the property that the preferred links are the shortest ones, and that the limit p → 0 is well-defined, as it corresponds to c(x) = log x, but has the disadvantage of having average cost −∞ when p ≤ −d).…”

“…This generic non-degeneracy property does not hold only for the d = p = 1 case (note that p = 1 is the exponent at which the cost function changes concavity, that is, the critical point for the ordered / non-crossing structural transition). It is known [4] that in this case there are (almost surely) at least two distinct optimal matchings for each instance of the problem, the ordered matching and the Dyck matching [5] which coincide only in the rather atypical case (with probability N !/(2N − 1)!!) in which, for all i, the i-th white and black points are consecutive along the segment.…”