Technical Note—A Polynomial Simplex Method for the Assignment Problem

Hung, Ming S.

doi:10.1287/opre.31.3.595

Cited by 59 publications

(36 citation statements)

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“…Competing methods, [1, 2,4,14,15,[18][19][20]23,24], including the Hungarian method, have complexity 0(;V''), so for large sparse problems the complexity of the auction algorithm is superior.…”

Section: Introductionmentioning

confidence: 99%

The auction algorithm for the transportation problem

1989

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The auction algorithm is a parallel relaxation nictliod for solving the classical assignment problem. It resembles a competitive bidding process whereby unussigncd persons bid simultaneously for objects, thereby raising their prices. Once all bids arc in, objects arc awarded to the highest bidder. Tliis paper generalizes the auction :ilgoritliin to solve linear transportation problems. The idea is to convert the transportation problem into an assignment problem, and then to inodily the auction algorithm to exploit the special structure of this problem. Computational results sliow Ihat this modified version of the auction algorithm is very efficient for certain types of transportation problems. IntroductionIn this paper, we propose a new relaxation algorithm for linear transportation problems. The algorithm resembles classical coordinate descent, Gauss-Seidel, and Jacobi methods for solving unconstrained nonlinear optimization problems or systems of nonlinear equations. It modifies the dual variables (node prices), either one at a time (Gauss-Seidel version) or all at once (Jacobi version) using only local node information, while aiming to improve the dual cost. It is well suited for implementation on massively parallel machines.The first relaxation algorithm for linear network flow problem was the auction algorithm for the classical assignment problem, proposed by the first author in 1979 [3] and further discussed in [8,13]. The algorithm operates like an auction whereby unassigned persons bid simultaneously for objects,, thereby raising their prices. Once all bids are in, objects are awarded to the highest bidder. The algorithm can also be interpreted as a Jacobi-like relaxation method for solving a dual problem. The variables of the dual problem may be viewed as the prices of the objects and are © J.C. Baltzer AG, Scientific Publishing Company

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Section: Introductionmentioning

confidence: 99%

The auction algorithm for the transportation problem

1989

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“…By combining these ideas with Dantzig's pivot rule (the new basic variable is that with the most negative reduced cost), Hung [104] obtained an algorithm which performs at most O(n 3 log∆) pivots, where ∆ is the difference between the objective function value of the starting solution with the optimal one. Orlin [132] reduced the number of pivots to O(n 3 log n) for the primal network simplex algorithm with Dantzig's rule.…”

Section: Primal Simplex-based Algorithmsmentioning

confidence: 99%

Linear Assignment Problems and Extensions

Burkard

Çela

1999

Handbook of Combinatorial Optimization

255

150

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This paper aims at describing the state of the art on linear assignment problems (LAPs). Besides sum LAPs it discusses also problems with other objective functions like the bottleneck LAP, the lexicographic LAP, and the more general algebraic LAP. We consider different aspects of assignment problems, starting with the assignment polytope and the relationship between assignment and matching problems, and focusing then on deterministic and randomized algorithms, parallel approaches, and the asymptotic behaviour. Further, we describe different applications of assignment problems, ranging from the well know personnel assignment or assignment of jobs to parallel machines, to less known applications, e.g. tracking of moving objects in the space. Finally, planar and axial three-dimensional assignment problems are considered, and polyhedral results, as well as algorithms for these problems or their special cases are discussed. The paper will appear in the Handbook of Combinatorial Optimization to be published by Kluwer Academic Publishers, P. Pardalos and D.-Z. Du, eds.

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“…It is of geometric although not of practical interest that this is better than the best previously known upper bound of n(n + 1)/2 steps, where each "step" represents a change either in the primal or the dual variables (see, e.g., Balinski and Gomory [1964], Munkres [1957]). The approach compares favorably both with Hung's [1983] recent primal simplex method for the assignment problem that generates at most n3 In A bases, where A is a constant that depends upon the costs, and with RoohyLaleh's [1981] primal simplex method that generates at most 0(n3) bases (independent of the costs).…”

Section: Believe It or Not Yet Another Approach To The Assignment Prmentioning

confidence: 99%

Signature Methods for the Assignment Problem

Balinski

1985

Operations Research

120

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Technical Note—A Polynomial Simplex Method for the Assignment Problem

Cited by 59 publications

References 0 publications

The auction algorithm for the transportation problem

The auction algorithm for the transportation problem

Linear Assignment Problems and Extensions

Signature Methods for the Assignment Problem

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