Finite-size corrections in the random assignment problem

Caracciolo, Sergio; D'Achille, Matteo; Malatesta, Enrico M.; Sicuro, Gabriele

doi:10.1103/physreve.95.052129

Cited by 9 publications

(13 citation statements)

References 32 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For this reason an analytical study of the finite-size corrections in the random TSP, where replicas are a natural tool, is still lacking. This is contrary to what happens to similar problems like the matching problem, where, in the random-link approximation these issues do not arise [9], and one can compute easily finite-size corrections [12,31], also for quite generic disorder distribution of the link variables [32].…”

Section: Introductionmentioning

confidence: 71%

Average optimal cost for the Euclidean TSP in one dimension

Caracciolo¹,

Gioacchino²,

Malatesta³

et al. 2019

J. Phys. A: Math. Theor.

Self Cite

View full text Add to dashboard Cite

The traveling-salesman problem is one of the most studied combinatorial optimization problems, because of the simplicity in its statement and the difficulty in its solution. We study the traveling salesman problem when the positions of the cities are chosen at random in the unit interval and the cost associated to the travel between two cities is their distance elevated to an arbitrary power p ∈ R. We characterize the optimal cycle and compute the average optimal cost for every number of cities when the measure used to choose the position of the cities is flat and asymptotically for large number of cities in the other cases. We also show that the optimal cost is a self-averaging quantity, and we test our analytical results with extensive simulations.Submitted to: J. Phys. A: Math. Gen.

show abstract

Section: Introductionmentioning

confidence: 71%

Average optimal cost for the Euclidean TSP in one dimension

Caracciolo¹,

Gioacchino²,

Malatesta³

et al. 2019

J. Phys. A: Math. Theor.

Self Cite

View full text Add to dashboard Cite

show abstract

“…In particular, any feasible configuration for the usual matching problem is feasible for the fractional matching problem; moreover, any feasible configuration for the fractional matching problem is feasible for the loopy fractional matching problem. and the finite-size corrections to the average optimal cost (aoc) [23,25] of the Rap have been extensively studied in the literature, and a closed formula is available for the aoc at any value of N in the case of exponentially distributed weights [12,13,26], namely…”

Section: Matchingmentioning

confidence: 99%

“…The expression above coincides with the one-site partition function in the Rmp and it has been discussed in details in Refs. [3,22,23,25]. In particular, Eqs.…”

Section: Appendix a One-site Partition Functionmentioning

confidence: 99%

The random fractional matching problem

Lucibello

Malatesta

Parisi³

et al. 2018

J. Stat. Mech.

Self Cite

View full text Add to dashboard Cite

We consider two formulations of the random-link fractional matching problem, a relaxed version of the more standard random-link (integer) matching problem. In one formulation, we allow each node to be linked to itself in the optimal matching configuration. In the other one, on the contrary, such a link is forbidden. Both problems have the same asymptotic average optimal cost of the random-link matching problem on the complete graph. Using a replica approach and previous results of Wästlund [1], we analytically derive the finitesize corrections to the asymptotic optimal cost. We compare our results with numerical simulations and we discuss the main differences between random-link fractional matching problems and the random-link matching problem.

show abstract

“…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 .…”

mentioning

confidence: 99%

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

Caracciolo¹,

D'Achille²,

Erba³

et al. 2020

J. Phys. A: Math. Theor.

Self Cite

View full text Add to dashboard Cite

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...

show abstract

Finite-size corrections in the random assignment problem

Cited by 9 publications

References 32 publications

Average optimal cost for the Euclidean TSP in one dimension

Average optimal cost for the Euclidean TSP in one dimension

The random fractional matching problem

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

Contact Info

Product

Resources

About