Asymptotic behavior of the expected optimal value of the multidimensional assignment problem

The Multidimensional Assignment Problem (MAP) is a higher-dimensional version of the Linear Assignment Problem that arises in the areas of data association, target tracking, resource allocation, etc. This paper elucidates the question of asymptotical behavior of the expected optimal value of the large-scale MAP whose assignment costs are independent identically distributed random variables with a prescribed probability distribution. We demonstrate that for a broad class of continuous distributions the limiting value of the expected optimal cost of the MAP is determined by the location of the left endpoint of the support set of the distribution, and construct asymptotical bounds for the expected optimal cost.

[1]  David E. Jeffcoat,et al.  APPLYING SIMULATED ANNEALING TO THE MULTIDIMENSIONAL ASSIGNMENT PROBLEM , 2004 .

[2]  Rainer E. Burkard,et al.  Selected topics on assignment problems , 2002, Discret. Appl. Math..

[3]  Carlos A. S. Oliveira,et al.  Asymptotic Properties of Random Multidimensional Assignment Problems , 2004 .

[4]  O. Barndorff-Nielsen,et al.  On the Limit Behaviour of Extreme Order Statistics , 1963 .

[5]  A. Erdélyi,et al.  General asymptotic expansions of laplace integrals , 1961 .

[6]  F. Olver.  Asymptotics and Special Functions , 1974 .

[7]  Egon Balas,et al.  An Algorithm for the Three-Index Assignment Problem , 1991, Oper. Res..

[8]  T. MacRobert.  Higher Transcendental Functions , 1955, Nature.

[9]  R. Karp.  An Upper Bound on the Expected Cost of an Optimal Assignment , 1987 .

[10]  B. Prabhakar,et al.  Proofs of the Parisi and Coppersmith‐Sorkin random assignment conjectures , 2005 .

[11]  Kenneth Steiglitz,et al.  Combinatorial Optimization: Algorithms and Complexity , 1981 .

[12]  Aubrey B. Poore,et al.  Data association problems posed as multidimensional assignment problems: problem formulation , 1993, Defense, Security, and Sensing.

[13]  B. Prabhakar,et al.  A Proof of the Conjecture due to Parisi for the Finite Random Assignment Problem , 2003 .

[14]  D. Aldous.  Asymptotics in the random assignment problem , 1992 .

[15]  William P. Pierskalla,et al.  Letter to the Editor - The Multidimensional Assignment Problem , 1968, Oper. Res..

[16]  David W. Walkup,et al.  On the Expected Value of a Random Assignment Problem , 1979, SIAM J. Comput..

[17]  Michel X. Goemans,et al.  A Lower Bound on the Expected Cost of an Optimal Assignment , 1993, Math. Oper. Res..

[18]  Rainer E. Burkard,et al.  Linear Assignment Problems and Extensions , 1999, Handbook of Combinatorial Optimization.

[19]  P. Pardalos,et al.  Handbook of Combinatorial Optimization , 1998 .

[20]  Aubrey B. Poore,et al.  A Numerical Study of Some Data Association Problems Arising in Multitarget Tracking , 1994 .

[21]  Cor J. Veenman,et al.  A fast and robust point tracking algorithm , 1998, Proceedings 1998 International Conference on Image Processing. ICIP98 (Cat. No.98CB36269).

[22]  Panos M. Pardalos,et al.  A greedy randomized adaptive search procedure for the multitarget multisensor tracking problem , 1997, Network Design: Connectivity and Facilities Location.

[23]  Mauro Dell'Amico,et al.  Annotated Bibliographies in Combinatorial Optimization , 1997 .

[24]  D. Aldous.  The ζ(2) limit in the random assignment problem , 2000, Random Struct. Algorithms.

[25]  Sharlene M. Andrijich,et al.  Solving the multisensor data association problem , 2001 .

[26]  W. Hager,et al.  Large Scale Optimization : State of the Art , 1993 .

[27]  Svante Linusson,et al.  A proof of Parisi’s conjecture on the random assignment problem , 2003, math/0303214.

[28]  Panos M. Pardalos,et al.  On the expected optimal value of random assignment problems: Experimental results and open questions , 1993, Comput. Optim. Appl..

[29]  Panos M. Pardalos,et al.  On the number of local minima for the multidimensional assignment problem , 2006, J. Comb. Optim..

[30]  M. Mézard,et al.  On the solution of the random link matching problems , 1987 .

[31]  Svante Linusson,et al.  A proof of a conjecture of Buck, Chan and Robbins on the random assignment problem , 2003 .

[32]  H. A. David,et al.  Order Statistics (2nd ed). , 1981 .

[33]  Panos M. Pardalos,et al.  Nonlinear assignment problems : algorithms and applications , 2000 .

[34]  Thomas M. Liebling,et al.  Tracking elementary particles near their primary vertex: A combinatorial approach , 1996, J. Glob. Optim..

[35]  G. Parisi.  A Conjecture on random bipartite matching , 1998, cond-mat/9801176.

[36]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[37]  M. Mézard,et al.  Replicas and optimization , 1985 .

[38]  Noga Alon,et al.  The Probabilistic Method, Second Edition , 2004 .

[39]  Hassen T. Dorrah,et al.  The multidimensional assignment problem with application , 1990, Proceedings of the 33rd Midwest Symposium on Circuits and Systems.

[40]  Panos M. Pardalos,et al.  GRASP with Path Relinking for Three-Index Assignment , 2005, INFORMS J. Comput..

[41]  Aubrey B. Poore,et al.  Multidimensional assignment formulation of data association problems arising from multitarget and multisensor tracking , 1994, Comput. Optim. Appl..

[42]  N. Alon,et al.  The Probabilistic Method, Second Edition , 2000 .

[43]  Andrew J. Lazarus,et al.  Certain expected values in the random assignment problem , 1993, Oper. Res. Lett..

[44]  Noga Alon,et al.  The Probabilistic Method , 2015, Fundamentals of Ramsey Theory.

[45]  Panos M. Pardalos,et al.  A Parallel Grasp for the Data Association Multidimensional Assignment Problem , 1999 .

[46]  Eduardo L. Pasiliao,et al.  Tree-Based Algorithms for the Multidimensional Assignment Problem , 2004 .