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Almost sure convergence of the minimum bipartite matching functional in Euclidean space [article]

J.H. Boutet de Monvel, O.C. Martin
2002 arXiv   pre-print
., Y_N) be the minimum length of a bipartite matching between two sets of points in R^d, where X_1,..., X_N,... and Y_1,..., Y_N,... are random points independently and uniformly distributed in [0,1]^d  ...  We prove that for d > 3, L_N/N^1-1/d converges with probability one to a constant β_MBM(d)>0 as N→∞ .  ... 
arXiv:math/0205140v1 fatcat:kjqhnjg77nbbdnjxbve7w7pkcq

Almost Sure Convergence of the Minimum Bipartite Matching Functional in Euclidean Space

J. H. Boutet de Monvel, O. C. Martin
2002 Combinatorica  
doi:10.1007/s00493-002-0004-x fatcat:uoeivqgdszeynptdvijbr4xlnu

Asymptotically Optimal Algorithms for One-to-One Pickup and Delivery Problems With Applications to Transportation Systems

Kyle Treleaven, Marco Pavone, Emilio Frazzoli
2013 IEEE Transactions on Automatic Control  
Our results leverage a novel connection between the Euclidean Bipartite Matching Problem and the theory of random permutations, and, for the dynamic setting, exhibit novel features that are absent in traditional  ...  First, by embedding the problem within a stochastic framework, we present a novel algorithm for the SCP that: (i) is asymptotically optimal, i.e., it produces, almost surely, a solution approaching the  ...  For any finite partition of , almost surely. Proof: Let denote the optimal bipartite matching of .  ... 
doi:10.1109/tac.2013.2259993 fatcat:nwc7ajvk2jhg3anmcwakvvjp5y

Cost bounds for Pickup and Delivery Problems with application to large-scale transportation systems

K. Treleaven, M. Pavone, E. Frazzoli
2012 2012 American Control Conference (ACC)  
The key strategy is to develop analytical bounds for the optimal cost of the Euclidean Stacker Crane Problem (ESCP), which represents a general static model for DRT systems.  ...  In this paper, our aim is to bridge this gap for a rather general model of DRT systems, which takes the form of a generalized Dynamic Pickup and Delivery Problem.  ...  ACKNOWLEDGMENTS The authors would like to thank Prof. Javed Aslam for suggesting the important connection between our lower bounds and the Wasserstain distance.  ... 
doi:10.1109/acc.2012.6315329 fatcat:rz4hyutym5fyhnrqv34isfi3ka

An asymptotically optimal algorithm for pickup and delivery problems

Kyle Treleaven, Marco Pavone, Emilio Frazzoli
2011 IEEE Conference on Decision and Control and European Control Conference  
Our results leverage a novel connection between the Euclidean Bipartite Matching Problem and the theory of random permutations.  ...  Pickup and delivery problems (PDPs), in which objects or people have to be transported between specific locations, are among the most common combinatorial problems in real-world operations.  ...  The Euclidean bipartite matching problem (EBMP), between sets X and Y of points in Euclidean space, is to find a permutation σ * (not necessarily unique) such that the sum of the Euclidean distances between  ... 
doi:10.1109/cdc.2011.6161406 dblp:conf/cdc/TreleavenPF11 fatcat:7qdjpiscm5h6hph4luyuwe4pdq

Page 2311 of Mathematical Reviews Vol. , Issue 2004c [page]

2004 Mathematical Reviews  
(F-PARIS11-TM; Orsay) Almost sure convergence of the minimum bipartite matching functional in Euclidean space. (English summary) Combinatorica 22 (2002), no. 4, 523-530.  ...  Summary: “Let Ly = Lygu(X%} Yy) be the min- imum length of a bipartite matching between two sets of points in R“, where X; a Yy,... are random points independently and uniformly distributed in [0, 1]/.  ... 

A Fast ℒpSpike Alignment Metric

Alexander J. Dubbs, Brad A. Seiler, Marcelo O. Magnasco
2010 Neural Computation  
The metrization of the space of neural responses is an ongoing research program seeking to find natural ways to describe, in geometrical terms, the sets of possible activities in the brain.  ...  We show how to implement a fast algorithm for the computation of this metric based on bipartite graph matching theory. * A. Dubbs is currently at the Mathematics Department, MIT.  ...  Kaiser and the RU Outreach program for support and encouragement. This work was supported in part by the National Science Foundation under grant EF-0928723.  ... 
doi:10.1162/neco_a_00026 pmid:20804383 fatcat:amdsateo4zcvpcvbz7yqejzesi

On Minimum Spanning Trees for Random Euclidean Bipartite Graphs [article]

Mario Correddu, Dario Trevisan
2021 arXiv   pre-print
We consider the minimum spanning tree problem on a weighted complete bipartite graph K_n_R, n_B whose n=n_R+n_B vertices are random, i.i.d. uniformly distributed points in the unit cube in d dimensions  ...  and edge weights are the p-th power of their Euclidean distance, with p>0.  ...  A full analysis was later performed by Steele [9] who proved that, if the Euclidean graph consists of n nodes, then for every 0 < p < d, almost sure convergence holds lim n→∞ C M ST (G n ) n 1−p/d =  ... 
arXiv:2107.08452v1 fatcat:7e222d6l6zgktkkv6blan5e5ze

Replica symmetry of the minimum matching

Johan Wästlund
2012 Annals of Mathematics  
Abstract We establish the soundness of the replica symmetric ansatz introduced by M. Mézard and G. Parisi for the minimum matching problem in the pseudo-dimension d mean field model for d ≥ 1.  ...  The case d = 1 corresponds to the π 2 /6-limit for the assignment problem proved by D. Aldous in 2001.  ...  I thank the referees for suggestions that greatly improved the presentation.  ... 
doi:10.4007/annals.2012.175.3.2 fatcat:wlsy5tqx5zaq5b6qkzm5jpkome

Replica Symmetry and Combinatorial Optimization [article]

Johan Wästlund
2009 arXiv   pre-print
Parisi for minimum matching and the traveling salesman problem in the pseudo-dimension d mean field model for d\geq 1.  ...  The case d=1 of minimum matching corresponds to the pi^2/6 limit for the assignment problem established by D.  ...  Almost sure uniqueness of valuation is equivalent to the statement that the sequence converges pointwise, and this in turn implies that F is the unique fixed point of W θ .  ... 
arXiv:0908.1920v2 fatcat:tnaq7upd45ahzel5b3vh46fb54

A new framework for Euclidean summary statistics in the neural spike train space

Sergiusz Wesolowski, Robert J. Contreras, Wei Wu
2015 Annals of Applied Statistics  
Statistical analysis and inference on spike trains is one of the central topics in the neural coding. It is of great interest to understand the underlying structure of given neural data.  ...  However, as those metrics lack certain Euclidean properties, the defined averages are nonunique, and do not share the conventional properties of a mean.  ...  Almost surely Non-unique Non-unique Not known Almost surely Table 1 we describe the schematic idea of incorporating the Isolated neuron responses (Not observed) Background noise (Observed  ... 
doi:10.1214/15-aoas847 fatcat:oj7kyuczwzh4hjidcl573hbr34

A Bayes consistent 1-NN classifier [article]

Aryeh Kontorovich, Roi Weiss
2018 arXiv   pre-print
We show that a simple modification of the 1-nearest neighbor classifier yields a strongly Bayes consistent learner.  ...  Prior to this work, the only strongly Bayes consistent proximity-based method was the k-nearest neighbor classifier, for k growing appropriately with sample size.  ...  The algorithm invokes a minimum vertex cover routine, which by König's theorem is equivalent to maximum matching for bipartite graphs, and is computable in randomized time O(n 2.376 ) (Mucha and Sankowski  ... 
arXiv:1407.0208v4 fatcat:i7khxho6nbc73a7g5v4oruiydu

Fundamental performance limits and efficient polices for Transportation-On-Demand systems

Marco Pavone, Kyle Treleaven, Emilio Frazzoli
2010 49th IEEE Conference on Decision and Control (CDC)  
as the inconvenience of privately-owned cars in metropolitan areas becomes excessive.  ...  Specifically, we study TOD systems in the form of a unit-capacity, multiple-vehicle dynamic pick-up and delivery problem, whereby pick-up requests arrive according to a Poisson process and are randomly  ...  Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of the supporting organizations.  ... 
doi:10.1109/cdc.2010.5717552 dblp:conf/cdc/PavoneTF10 fatcat:trs52kaftngabpaql27teqd5zi

Limit theory of combinatorial optimization for random geometric graphs [article]

Dieter Mitsche, Mathew D. Penrose
2020 arXiv   pre-print
In the random geometric graph G(n,r_n), n vertices are placed randomly in Euclidean d-space and edges are added between any pair of vertices distant at most r_n from each other.  ...  , bipartite matching and bipartite travelling salesman problems, for a general class of weight functions with at most polynomial growth of order d-ε, under thermodynamic scaling of the distance parameter  ...  Here we obtain LLNs under thermodynamic scaling for a general class of weight functions f , not only for the TSP but also for the minimum-weight matching (MM), minimum spanning tree (MST) and minimum bipartite  ... 
arXiv:2006.14915v1 fatcat:uk7mnj3fxjctpat3u5kc6pn334

Methods for computing state similarity in Markov Decision Processes [article]

Norman Ferns, Pablo Samuel Castro, Doina Precup, Prakash Panangaden
2012 arXiv   pre-print
We obtain in this manner a variety of distance functions for MDP state aggregation, which differ in the tradeoff between time and space complexity, as well as the quality of the aggregation.  ...  A popular approach to solving large probabilistic systems relies on aggregating states based on a measure of similarity. Many approaches in the literature are heuristic.  ...  Acknowledgments This work has been supported in part by funding from NSERC and CFI.  ... 
arXiv:1206.6836v1 fatcat:kpl4f6g7sfbrpkudsfianqqoue
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