Quadratic Stochastic Euclidean Bipartite Matching Problem

Caracciolo, Sergio; Sicuro, Gabriele

doi:10.1103/physrevlett.115.230601

Cited by 23 publications

(59 citation statements)

References 24 publications

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“…where a(N ) b(N ) if a(N ) b(N ) tends to a non-zero finite constant for N → ∞. In this case a relation with the classical optimal transport problem in the continuum has been exploited [13], in particular for the case p = 2 where very detailed results have been obtained [14][15][16].…”

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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.

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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...

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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.

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“…However, it is not clear in general how to apply these techniques (beyond expanding around the mean field case [32,33]), when correlations play an important role, as happens when the graph is embedded in Euclidean spaces. For other problems besides the TSP, analysis of the one-dimensional case has enabled progress in the study of higher-dimensional cases [34]. As a consequence, a relevant question is whether the relations we obtained in one dimension continue to exist also in d > 1, where the bipartite TSP is an NP-complete problem.…”

Section: Discussionmentioning

confidence: 94%

Solution for a bipartite Euclidean traveling-salesman problem in one dimension

Caracciolo¹,

Gioacchino²,

Malatesta³

2018

Phys. Rev. E

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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 characterize the optimal cycle for every convex and increasing cost function when the points are thrown independently and with an identical probability distribution in a compact interval. We compute the average optimal cost for every number of points when the distance function is the square of the Euclidean distance. We also show that the average optimal cost is not a self-averaging quantity by explicitly computing the variance of its distribution in the thermodynamic limit. Moreover, we prove that the cost of the optimal cycle is not smaller than twice the cost of the optimal assignment of the same set of points. Interestingly, this bound is saturated in the thermodynamic limit.

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“…[2] the scaling of the AOC in the REAP with weight function w p on the unit square has been derived, whereas in Ref. [3,4] the coefficient to this scaling has been obtained for p = 2 both on the unit square and on the torus, a result later rigorously proved in Ref. [5].…”

Section: Introductionmentioning

confidence: 98%

“…[4,20], the provided general formula predict ε N = O N − p /2 but it may present some issues, in particular can be divergent depending on the properties of . In this case, the expression must be properly regularized, and a different, anomalous asymptotic behavior of ε N may appear [4]. Here we will present a re-derivation of such a general expression for the AOC in the REAP on the line, extending the arguments in Refs.…”

Section: Introductionmentioning

confidence: 99%

Anomalous Scaling of the Optimal Cost in the One-Dimensional Random Assignment Problem

2018

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We consider the random Euclidean assignment problem on the line between two sets of N random points, independently generated with the same probability density function . The cost of the matching is supposed to be dependent on a power p > 1 of the Euclidean distance of the matched pairs. We discuss an integral expression for the average optimal cost for N 1 that generalizes a previous result obtained for p = 2. We also study the possible divergence of the given expression due to the vanishing of the probability density function. The provided regularization recipe allows us to recover the proper scaling law for the cost in the divergent cases, and possibly some of the involved coefficients. The possibility that the support of is a disconnected interval is also analysed. We exemplify the proposed procedure and we compare our predictions with the results of numerical simulations.

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Quadratic Stochastic Euclidean Bipartite Matching Problem

Cited by 23 publications

References 24 publications

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

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

Solution for a bipartite Euclidean traveling-salesman problem in one dimension

Anomalous Scaling of the Optimal Cost in the One-Dimensional Random Assignment Problem

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