Solving the Many to Many assignment problem by improving the Kuhn-Munkres algorithm with backtracking

The Many to Many (M-M) assignment problem is an important open problem where one task is assigned to many, but different, agents and one agent may undertake many, but different, tasks. The Kuhn-Munkres (K-M) algorithm is a famous and traditional process used in dealing with assignment problems. In this paper, we propose a solution to the M-M assignment problem by improving the K-M algorithm with backtracking (KMB). To demonstrate the solution's suitability, we prove that the proposed KMB algorithm is valid and that the worst time complexity of the KMB algorithm is O ( ( ? L a i ) 3 ) , where L a i denotes the maximum number of tasks that can be assigned to agent i. After that, we discuss several critical problems related to the algorithm and provide the necessary and sufficient conditions of solving the M-M assignment problem. Finally, we demonstrate, through experimentation, the validity, practicality and efficiency of the KMB algorithm. It improves the K-M algorithm to solve the M-M assignment problem.The improved algorithm (KMB) introduces backtracking.The KMB algorithm is valid and the worst time complexity is O ( ( ? L a i ) 3 ) .It provides the necessary and sufficient conditions for the solution.It illustrates the validity and efficiency of the KMB algorithm through simulations.

[1]  François Bourgeois,et al.  An extension of the Munkres algorithm for the assignment problem to rectangular matrices , 1971, CACM.

[2]  Mauro Dell'Amico,et al.  8. Quadratic Assignment Problems: Algorithms , 2009 .

[3]  Ismail Hakki Toroslu,et al.  Incremental assignment problem , 2007, Inf. Sci..

[4]  H. Van Dyke Parunak,et al.  Changing Roles: Dynamic Role Assignment , 2003, J. Object Technol..

[5]  Manuela Veloso,et al.  Multi-Robot Dynamic Role Assignment and Coordination Through Shared Potential Fields , 2002 .

[6]  J. Munkres.  ALGORITHMS FOR THE ASSIGNMENT AND TRANSIORTATION tROBLEMS* , 1957 .

[7]  Edmund H. Durfee,et al.  Using hybrid scheduling for the semi-autonomous formation of expert teams , 2014, Future Gener. Comput. Syst..

[8]  Edwin R. Hancock,et al.  Graph matching using the interference of continuous-time quantum walks , 2009, Pattern Recognit..

[9]  Frank Dignum,et al.  Role-assignment in open agent societies , 2003, AAMAS '03.

[10]  András Frank,et al.  On Kuhn's Hungarian Method—A tribute from Hungary , 2005 .

[11]  MengChu Zhou,et al.  Group Role Assignment via a Kuhn–Munkres Algorithm-Based Solution , 2012, IEEE Transactions on Systems, Man, and Cybernetics - Part A: Systems and Humans.

[12]  Mauro Dell'Amico,et al.  The k-cardinality Assignment Problem , 1997, Discret. Appl. Math..

[13]  Manuela M. Veloso,et al.  Task Decomposition, Dynamic Role Assignment, and Low-Bandwidth Communication for Real-Time Strategic Teamwork , 1999, Artif. Intell..

[14]  Anantha Chandrakasan,et al.  Bounding the lifetime of sensor networks via optimal role assignments , 2002, Proceedings.Twenty-First Annual Joint Conference of the IEEE Computer and Communications Societies.

[15]  Jacques Ferber,et al.  From Agents to Organizations: An Organizational View of Multi-agent Systems , 2003, AOSE.

[16]  H. Kuhn.  The Hungarian method for the assignment problem , 1955 .

[17]  Mauro Dell'Amico,et al.  Assignment Problems , 1998, IFIP Congress: Fundamentals - Foundations of Computer Science.

[18]  Dong Xu,et al.  Face and Human Gait Recognition Using Image-to-Class Distance , 2010, IEEE Transactions on Circuits and Systems for Video Technology.

[19]  W. Brogan.  Algorithm for Ranked Assignments with Applications to Multiobject Tracking , 1989 .

[20]  Narendra Karmarkar,et al.  A new polynomial-time algorithm for linear programming , 1984, Comb..

[21]  Young-June Choi,et al.  Multichannel wireless scheduling under limited terminal capability , 2008, IEEE Transactions on Wireless Communications.

[22]  MengChu Zhou,et al.  M-M Role-Transfer Problems and Their Solutions , 2009, IEEE Trans. Syst. Man Cybern. Part A.

[23]  Gwo-Hshiung Tzeng,et al.  Multi-criteria task assignment in workflow management systems , 2003, 36th Annual Hawaii International Conference on System Sciences, 2003. Proceedings of the.

[24]  Dimitri P. Bertsekas,et al.  Parallel synchronous and asynchronous implementations of the auction algorithm , 1991, Parallel Comput..