The Optimal Assignment Problem: An Investigation Into Current Solutions, New Approaches and the Doubly Stochastic Polytope Frans-Willem Vermaak A Dissertation submitted to the Faculty of Engineering, University of the Witwatersand, Johannesburg, in fulfilment of the requirements for the degree of Master of Science in Engineering Johannesburg 2010 Declaration I declare that this research dissertation is my own, unaided work. It is being submitted for the Degree of Master of Science in Engineering in the University of the Witwatersrand, Johannesburg. It has not been submitted before for any degree or examination in any other University. signed this day of 2010 ii Abstract This dissertation presents two important results: a novel algorithm that approximately solves the optimal assignment problem as well as a novel method of projecting matrices into the doubly stochastic polytope while preserving the optimal assignment. The optimal assignment problem is a classical combinatorial optimisation problem that has fuelled extensive research in the last century. The problem is concerned with a matching or assignment of elements in one set to those in another set in an optimal manner. It finds typical application in logistical optimisation such as the matching of operators and machines but there are numerous other applications. In this document a process of iterative weighted normalization applied to the benefit matrix associated with the Assignment problem is considered. This process is derived from the application of the Computational Ecology Model to the assignment problem and referred to as the OACE (Optimal Assignment by Computational Ecology) algorithm. This simple process of iterative weighted normalisation converges towards a matrix that is easily converted to a permutation matrix corresponding to the optimal assignment or an assignment close to optimality. The document also considers a method of projecting a matrix into the doubly stochastic polytope while preserving the optimal assignment. Various methods of projecting square matrices into the doubly stochastic polytope exist but none that preserve the assignment. This novel result could prove instrumental in solving assignment problems and promises applications in other optimisation algorithms similar to those that Sinkhorn’s algorithm finds. iii CONTENTS Chapter 1: Introduction .......................................................................................................................................................... 1 1.1 Introduction ............................................................................................................................................................. 1 1.2 Problem statement ............................................................................................................................................... 3 Chapter2: Literature survey ................................................................................................................................................. 4 2.1 General concepts .................................................................................................................................................... 4 2.1.1 Permutation matrices ............................................................................................................................... 4 2.1.2 Doubly stochastic matrices .................................................................................................................... 5 2.1.3 The assignment problem ......................................................................................................................... 7 2.2 Applications of the assignment problem .................................................................................................. 9 2.2.1 Reformulating the shortest path problem as an assignment problem............................ 9 2.3 Methods of solving the assignment problem ....................................................................................... 10 2.3.1 The assignment problem considered as a linear program ................................................. 10 2.3.2 The Dual Problem .................................................................................................................................... 11 2.3.3 Hungarian method ................................................................................................................................... 12 2.3.4 Auction algorithm .................................................................................................................................... 19 2.3.5 Invisible hand algorithm: Solving the assignment problem using a statistical physics approach .......................................................................................................................................................... 24 2.4 Other Important Concepts ............................................................................................................................. 30 2.4.1 Hogg and Huberman’s Computational Ecology ........................................................................ 30 Chapter 3: The OACE Algorithm...................................................................................................................................... 36 3.1 Introduction .......................................................................................................................................................... 36 3.2 Application of the Computational Ecology structure to the assignment problem ........... 37 3.3 OACE algorithm ................................................................................................................................................... 39 3.3.1 Initialisation of the algorithm ............................................................................................................ 40 3.3.2 Termination of the algorithm ............................................................................................................ 40 3.3.3 OACE Algorithm viewed as a process of iterative weighted normalisation .............. 42 3.3.4 Convergence of the population matrix .......................................................................................... 43 3.3.5 Results ............................................................................................................................................................ 46 iv 3.3.6 Simulation results .................................................................................................................................... 46 3.3.7 Computational efficiency of the algorithm.................................................................................. 49 3.3.8 Examples and applications of the OACE algorithm ................................................................ 54 3.4 Comparison of the OACE algorithm to other algorithms ............................................................... 59 3.4.1 Brute force approach.............................................................................................................................. 59 3.4.2 OACE algorithm ......................................................................................................................................... 60 3.4.3 Auction algorithm .................................................................................................................................... 60 3.4.4 Hungarian algorithm .............................................................................................................................. 60 3.4.5 Graphical comparisons of the computational complexity of the different algorithms ........................................................................................................................................................................ 61 3.4.6 Tabled Summary of comparisons .................................................................................................... 63 Chapter 4: Doubly stochastic approximation of a matrix with the same optimal assignment....... 64 4.1 Introduction .......................................................................................................................................................... 64 4.2 Derivation ............................................................................................................................................................... 64 4.3 Computational complexity of algorithm ................................................................................................. 71 4.4 Example ................................................................................................................................................................... 72 4.5 Discussion of assignment preserving projection method ............................................................. 74 Chapter 5 ..................................................................................................................................................................................... 75 5.1 Conclusion .............................................................................................................................................................. 75 5.2 Future work ........................................................................................................................................................... 76 5.2.1 Analytical derivation of OACE algorithm ....................................................................................