We study the random-link matching problem on random regular graphs, alongside with two relaxed versions of the problem, namely the fractional matching and the so-called "loopy" fractional matching. We estimated the asymptotic average optimal cost using the cavity method. Moreover, we also study the finite-size corrections due to rare topological structures appearing in the graph at large sizes. We estimate these contributions using the cavity approach, and we compare our results with the output of numerical simulations. The analysis also clarifies the meaning of the finite-size contributions appearing in the fullyconnected version of the problem, that has been already analyzed in the literature.Random-link matching problems on random regular graphs 2 relevance. In particular, the theory of disordered systems seems to be especially suitable for the study of the typical properties of optimization problems in presence of randomness [8][9][10][11]. The exploration of this topic, that also inspired powerful numerical techniques for the algorithmic solution of these problems, started in the eighties, with the seminal works of Orland [12] and Mézard and Parisi [13], that tackled, as prototype problem, the random-link matching problem on the complete graph. They proved that the random-link matching problem can be fully investigated using the replica method and the cavity method in the limit of large number of vertices [14]. Exact results have been obtained about the leading order cost and the finite-size corrections [15][16][17], the fluctuation of the average optimal cost [18], and its embedding in the Euclidean space [19][20][21].Despite its popularity in the fully-connected version, very few results are available on the random-link matching problem on sparse topologies. Zhou and Ou-Yang [22] and subsequently Zdeborová and Mézard [23] studied, using the cavity method, the number of maximum and perfect (unweighted) matchings on sparse graphs. As far as we know, a study of the random weighted matching problem on the Bethe lattice is, instead, missing in the literature.In this paper, we investigate exactly this formulation of the matching problem using the cavity method, both at the leading order and at the level of finite-size corrections. This analysis will be, first of all, of methodological interest: indeed, we will check, for the first time to our knowledge, the effectiveness of the cavity method in the evaluation of the contribution to finitesize corrections of rare topological structures in a combinatorial optimization problem defined on a sparse graph. Such an approach, introduced in the study of spin glasses in Refs. [24][25][26], allows us to go beyond the leading order using results (like the cavity fields distribution) obtained at the leading level. Moreover, these topological contributions seem to be related to the finitesize corrections appearing in the fully-connected case, a fact suggested in Ref.[17] that will be clarified in the present paper, where this correspondence will be made explicit.The paper is ...
