Multi index assignment problems: complexity, approximation, applications

This chapter deals with approximation algorithms for and applications of multi index assignment problems (MIAPs). MIAPs and relatives of it have a relatively long history both in applications as well as in theoretical results, starting at least in the fifties (see e.g. [Motzkin, 1952], [Schell, 1955] and [Koopmans and Beckmann, 1957]). Here we intend to give the reader i) an idea of the range and diversity of practical problems that have been formulated as an MIAP, and ii) an overview on what is known on theoretical aspects of solving instances of MIAPs. In particular, we will discuss complexity and approximability issues for special cases of MIAPs. We feel that investigating special cases of MIAPs is an important topic since real-world instances almost always posses a certain structure that can be exploited when it comes to solving them.

[1]  M. N. Vartak,et al.  Time-cost tradeoff in a three dimensional assignment problem , 1989 .

[2]  Aubrey B. Poore,et al.  A Lagrangian Relaxation Algorithm for Multidimensional Assignment Problems Arising from Multitarget Tracking , 1993, SIAM J. Optim..

[3]  Viggo Kann,et al.  Maximum Bounded 3-Dimensional Matching is MAX SNP-Complete , 1991, Inf. Process. Lett..

[4]  P. Miliotis,et al.  An algorithm for the planar three-index assignment problem , 1994 .

[5]  W. Oettli,et al.  Advances in Optimization , 1992 .

[6]  Takao Nishizeki,et al.  Planar Graphs: Theory and Algorithms , 1988 .

[7]  Rainer E. Burkard,et al.  On latin squares and the facial structure of related polytopes , 1986, Discret. Math..

[8]  M. C. Puri,et al.  The three dimensional bottleneck assignment problem and its variants , 1985 .

[9]  Yves Crama,et al.  Approximation algorithms for three-dimensional assignment problems with triangle inequalities , 1992 .

[10]  Egon Balas,et al.  Linear-Time Separation Algorithms for the Three-Index Assignment Polytope , 1993, Discret. Appl. Math..

[11]  James R. Evans.  The multicommodity assignment problem: a network aggregation heuristic , 1981 .

[12]  Martin E. Dyer,et al.  Planar 3DM is NP-Complete , 1986, J. Algorithms.

[13]  Kenneth C. Gilbert,et al.  An Algorithm for a Class of Three-Dimensional Assignment Problems Arising in Scheduling Applications , 1987 .

[14]  R. Burkard,et al.  Computational investigations on 3-dimensional axial assignment problems , 1993 .

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

[16]  Egon Balas,et al.  Facets of the three-index assignment polytope , 1989, Discret. Appl. Math..

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

[18]  Dario Pacciarelli,et al.  A Three-dimensional Matching Model for Perishable Production Scheduling , 1999, Discret. Appl. Math..

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

[20]  Frits C. R. Spieksma,et al.  Production planning in automated manufacturing , 1994 .

[21]  Frits C. R. Spieksma,et al.  Approximation Algorithms for Multi-Dimensional Assignment Problems with Decomposable Costs , 1994, Discret. Appl. Math..

[22]  Michael Jünger,et al.  Operations Research ’93 , 1994 .

[23]  Raymond E. Miller,et al.  Complexity of Computer Computations , 1972 .

[24]  M. N. Vartak,et al.  The three-dimensional bottleneck assignment problem with capacity constraints , 1994 .

[25]  Liqun Qi,et al.  On facets of the three-index assignment polytope , 1992, Australas. J Comb..

[26]  Esther M. Arkin,et al.  On Local Search for Weighted k-Set Packing , 1998, Math. Oper. Res..

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

[28]  D. Magos,et al.  Tabu Search for the planar three-index assignment problem , 1996, J. Glob. Optim..

[29]  D. Fortin,et al.  Routing in Meshes Using Linear Assignment Problem , 1994 .

[30]  A. Frieze.  Complexity of a 3-dimensional assignment problem , 1983 .

[31]  L. Qi,et al.  A NEW FACET CLASS AND A POLYHEDRAL METHOD FOR THE THREE-INDEX ASSIGNMENT PROBLEM , 1994 .

[32]  Kenneth C. Gilbert,et al.  MULTIDIMENSIONAL ASSIGNMENT PROBLEMS , 1988 .

[33]  Egon Balas,et al.  Traffic assignment in communication satellites , 1983 .

[34]  A. J. W. Hilton.  The reconstruction of latin squares with applications to school timetabling and to experimental design , 1980 .

[35]  T. Koopmans,et al.  Assignment Problems and the Location of Economic Activities , 1957 .

[36]  Gerhard J. Woeginger,et al.  Three-dimensional Axial Assignment Problems with Decomposable Cost Coefficients , 1996, Discret. Appl. Math..

[37]  Alexander Schrijver,et al.  On the Size of Systems of Sets Every t of Which Have an SDR, with an Application to the Worst-Case Ratio of Heuristics for Packing Problems , 1989, SIAM J. Discret. Math..

[38]  Y. Bar-Shalom,et al.  A new relaxation algorithm and passive sensor data association , 1992 .

[39]  F. Spieksma,et al.  Geometric three-dimensional assignment problems☆ , 1996 .

[40]  Gerhard J. Woeginger,et al.  A New Efficiently Solvable Special Case of the Three-Dimensional Axial Bottleneck Assignment Problem , 1995, Combinatorics and Computer Science.

[41]  B. Korte,et al.  Worst case analysis of greedy type algorithms for independence systems , 1980 .