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Journal Article•DOI•

A competitive (dual) simplex method for the assignment problem

Michel Balinski1•
École Polytechnique1
01 Mar 1986-Mathematical Programming (Springer-Verlag New York, Inc.)-Vol. 34, Iss: 2, pp 125-141
TL;DR: A dual simplex method for the assignment problem leaves open to choice the activity (i,j) of rowi and columnj that is to be dropped in pivoting so long asxij < 0.1, and it is argued that on average the number of pivots is at mostn logn.
Abstract: “Where there is abundance of mystery and confusion in every direction, the truth seldom remains hidden for long. It's a matter of having plenty of angles to go at it from. Only the utterly simple crimes - the simplex crimes, you may say - have the trick of remaining baffling.” - Sir John (from Michael Innes,The Open House (A Sir John Appleby Mystery), Penguin Books, 1974).
Citations
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Journal Article•DOI•
A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem

[...]

Jean-David Benamou1, Yann Brenier2•
French Institute for Research in Computer Science and Automation1, Institut Universitaire de France2
01 Jan 2000-Numerische Mathematik
TL;DR: The Monge-Kantorovich mass transfer problem is reset in a fluid mechanics framework and numerically solved by an augmented Lagrangian method.
Abstract: Summary. The $L^2$ Monge-Kantorovich mass transfer problem [31] is reset in a fluid mechanics framework and numerically solved by an augmented Lagrangian method.

1,573 citations

Cites methods from "A competitive (dual) simplex method..."

  • ...There are nearly optimal algorithms for general cost matrices, with a computational cost of order O(N2 log N), such as Balinski’s algorithm [2]....

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Book•
Network Optimization: Continuous and Discrete Models

[...]

Dimitri P. Bertsekas
01 May 1998

874 citations

Journal Article•DOI•
The auction algorithm: a distributed relaxation method for the assignment problem

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Dimitri P. Bertsekas1•
Massachusetts Institute of Technology1
01 Jun 1988-Annals of Operations Research
TL;DR: A massively parallelizable algorithm for the classical assignment problem was proposed in this article, where unassigned persons bid simultaneously for objects thereby raising their prices. Once all bids are in, objects are awarded to the highest bidder.
Abstract: We propose a massively parallelizable algorithm for the classical assignment problem. 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. Its (sequential) worst — case complexity, for a particular implementation that uses scaling, is O(NAlog(NC)), where N is the number of persons, A is the number of pairs of persons and objects that can be assigned to each other, and C is the maximum absolute object value. Computational results show that, for large problems, the algorithm is competitive with existing methods even without the benefit of parallelism. When executed on a parallel machine, the algorithm exhibits substantial speedup.

649 citations

Cites methods from "A competitive (dual) simplex method..."

  • ...Among the many methods for the assignment problem [11] -[25], the auction algorithm seems to be the only one that has a naturally parallel character and is well suited for implementation on a massively parallel machine....

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Book Chapter•DOI•
Linear Assignment Problems and Extensions

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Rainer E. Burkard1, Eranda Çela1•
University of Graz1
01 Jan 1999
TL;DR: Assignment problems deal with the question how to assign n items to n machines (or workers) in the best possible way and an objective function modeling the ”best way” is modeled.
Abstract: Assignment problems deal with the question how to assign n items (eg jobs) to n machines (or workers) in the best possible way They consist of two components: the assignment as underlying combinatorial structure and an objective function modeling the ”best way”

344 citations

Cites methods from "A competitive (dual) simplex method..."

  • ...Kleinschmidt, Lee, and Schannath [114] have shown that this algorithm of Balinski [19] is equivalent to an algorithm proposed earlier by Hung and Rom [105]....

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  • ...A similar approach using the framework of strong spanning trees (see also Section 3 and Balinski and Gonzalez [21]) is given by Armstrong and Jin [10]....

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  • ...Paparrizos’ algorithm is similar to Balinski’s competitive simplex algorithm [19]....

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  • ...The algorithm of Balinski and Gomory is probably the oldest primal algorithm for the LSAP....

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  • ...The algorithm starts with a Balinski tree....

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Journal Article•DOI•
Adaptivity with moving grids

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Chris Budd1, Weizhang Huang2, Robert D. Russell3•
University of Bath1, University of Kansas2, Simon Fraser University3
01 May 2009-Acta Numerica
TL;DR: R-adaptive methods have enormous potential and indeed can produce an optimal form of adaptivity for many problems, including scale-invariant problems, blow-up problems, problems with moving fronts and problems in meteorology.
Abstract: In this article we look at the modern theory of moving meshes as part of an r-adaptive strategy for solving partial differential equations with evolving internal structure. We firstly examine the possible geometries of a moving mesh in both one and higher dimensions, and the discretization of partial differential equation on such meshes. In particular, we consider such issues as mesh regularity, equidistribution, variational methods, and the error in interpolating a function or truncation error on such a mesh. We show that, guided by these, we can design effective moving mesh strategies. We then look in more detail as to how these strategies are implemented. Firstly we look at position-based methods and the use of moving mesh partial differential equation (MMPDE), variational and optimal transport methods. This is followed by an analysis of velocity-based methods such as the geometric conservation law (GCL) methods. Finally we look at a number of examples where the use of a moving mesh method is effective in applications. These include scale-invariant problems, blow-up problems, problems with moving fronts and problems in meteorology. We conclude that, whilst r-adaptive methods are still in a relatively new stage of development, with many outstanding questions remaining, they have enormous potential for development, and for many problems they represent an optimal form of adaptivity.

277 citations

Cites background or methods from "A competitive (dual) simplex method..."

  • ...Robust methods exist for solving the corresponding linear programming problem, but to the best of our knowledge these methods typically require O(N2) operations (Kaijser 1998, Balinski 1986), which is unacceptable except for small problems....

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  • ...So far a number of moving mesh methods and a variety of variants have been developed and successfully applied to practical problems; see the review articles of Cao et al. (2003), Eisman (1985, 1987), Hawken, Gottlieb and Hansen (1991), Thompson (1985), Thompson, Warsi and Mastin (1982) and Thompson and Weatherill (1992), and the books of Baines (1994), Carey (1997), Knupp and Steinberg (1994), Liseikin (1999), Thompson, Warsi and Mastin (1985) and Zegeling (1993)....

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  • ...(2003), Eisman (1985, 1987), Hawken, Gottlieb and Hansen (1991), Thompson (1985), Thompson, Warsi and Mastin (1982) and Thompson and Weatherill (1992), and the books of Baines (1994), Carey (1997), Knupp and Steinberg (1994), Liseikin (1999), Thompson, Warsi and Mastin (1985) and Zegeling (1993)....

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References
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Journal Article•DOI•
Theoretical Improvements in Algorithmic Efficiency for Network Flow Problems

[...]

Jack Edmonds1, Richard M. Karp2•
University of Waterloo1, University of California, Berkeley2
01 Apr 1972-Journal of the ACM
TL;DR: New algorithms for the maximum flow problem, the Hitchcock transportation problem, and the general minimum-cost flow problem are presented, and Dinic shows that, in a network with n nodes and p arcs, a maximum flow can be computed in 0 (n2p) primitive operations by an algorithm which augments along shortest augmenting paths.
Abstract: This paper presents new algorithms for the maximum flow problem, the Hitchcock transportation problem, and the general minimum-cost flow problem. Upper bounds on the numbers of steps in these algorithms are derived, and are shown to compale favorably with upper bounds on the numbers of steps required by earlier algorithms. First, the paper states the maximum flow problem, gives the Ford-Fulkerson labeling method for its solution, and points out that an improper choice of flow augmenting paths can lead to severe computational difficulties. Then rules of choice that avoid these difficulties are given. We show that, if each flow augmentation is made along an augmenting path having a minimum number of arcs, then a maximum flow in an n-node network will be obtained after no more than ~(n a - n) augmentations; and then we show that if each flow change is chosen to produce a maximum increase in the flow value then, provided the capacities are integral, a maximum flow will be determined within at most 1 + logM/(M--1) if(t, S) augmentations, wheref*(t, s) is the value of the maximum flow and M is the maximum number of arcs across a cut. Next a new algorithm is given for the minimum-cost flow problem, in which all shortest-path computations are performed on networks with all weights nonnegative. In particular, this algorithm solves the n X n assigmnent problem in O(n 3) steps. Following that we explore a "scaling" technique for solving a minimum-cost flow problem by treating a sequence of derived problems with "scaled down" capacities. It is shown that, using this technique, the solution of a Iiitchcock transportation problem with m sources and n sinks, m ~ n, and maximum flow B, requires at most (n + 2) log2 (B/n) flow augmentations. Similar results are also given for the general minimum-cost flow problem. An abstract stating the main results of the present paper was presented at the Calgary International Conference on Combinatorial Structures and Their Applications, June 1969. In a paper by l)inic (1970) a result closely related to the main result of Section 1.2 is obtained. Dinic shows that, in a network with n nodes and p arcs, a maximum flow can be computed in 0 (n2p) primitive operations by an algorithm which augments along shortest augmenting paths. KEY WOl¢l)S AND PHP~ASES: network flows, transportation problem, analysis of algorithms CR CATEGOI{.IES: 5.3, 5.4, 8.3

2,186 citations

"A competitive (dual) simplex method..." refers methods in this paper

  • ...This is competitive with the Edmonds-Karp specialization of the Hungarian method [ 8 ]....

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Theoretical Improvements in Algorithmic Efficiency for Network Flow Problems.

[...]

Jack Edmonds1, Richard M. Karp2•
National Institute of Standards and Technology1, University of California, Berkeley2
01 Jan 2001
TL;DR: In this article, the authors presented new algorithms for the maximum flow problem, the Hitchcock transportation problem and the general minimum-cost flow problem and derived upper bounds on the number of steps in these algorithms.
Abstract: This paper presents new algorithms for the maximum flow problem, the Hitchcock transportation problem and the general minimum-cost flow problem. Upper bounds on the number of steps in these algorithms are derived, and are shown to improve on the upper bounds of earlier algorithms.

2,081 citations

Journal Article•DOI•
A network simplex method

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William H. Cunningham1•
Johns Hopkins University1
01 Dec 1976-Mathematical Programming
TL;DR: Simple combinatorial modifications are given which ensure finiteness in the primal simplex method for the transshipment problem and the upper-bounded primalsimplexmethod for the minimum cost flow problem.
Abstract: Simple combinatorial modifications are given which ensure finiteness in the primal simplex method for the transshipment problem and the upper-bounded primal simplex method for the minimum cost flow problem. The modifications involve keeping "strongly feasible" bases. An efficient algorithm is given for converting any feasible basis into a strongly feasible basis. Strong feasibility is preserved by a rule for choosing the leaving basic variable at each simplex iteration. The method presented is closely related to a new perturbation technique and to previously known degeneracy modifications for shortest path problems and maximum flow problems.

195 citations

"A competitive (dual) simplex method..." refers background in this paper

  • ...The 'strongly feasible bases' of networks introduced by Cunningham [ 6 , 7] are, in the context of the assignment problem, bases with this same signature....

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Journal Article•DOI•
The alternating basis algorithm for assignment problems

[...]

Richard S. Barr1, Fred Glover2, Darwin Klingman3•
Southern Methodist University1, University of Colorado Boulder2, University of Texas at Austin3
01 Dec 1977-Mathematical Programming
TL;DR: A new primal extreme point algorithm for solving assignment problems which both circumvents and exploits degeneracy is presented, and is substantially more efficient than previously developed primal and primal-dual extreme point methods for assignment problems.
Abstract: The purpose of this paper is to present a new primal extreme point algorithm for solving assignment problems which both circumvents and exploits degeneracy. The algorithm is based on the observation that the degeneracy difficulties of the simplex method result from the unnecessary inspection of alternative basis representations of the extreme points. This paper characterizes a subsetQ of all bases that are capable of leading to an optimal solution to the problem if one exists. Using this characterization, an extreme point algorithm is developed which considers only those bases inQ. Computational results disclose that the new algorithm is substantially more efficient than previously developed primal and primal-dual extreme point (“simplex”) methods for assignment problems.

147 citations

"A competitive (dual) simplex method..." refers methods in this paper

  • ...The 'alternating bases' algorithm of Barr, Glover and Klingman [ 5 ] is a simplex method that pivots to preserve this signature....

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Journal Article•DOI•
Signature Methods for the Assignment Problem

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Michel Balinski1•
École Polytechnique1
01 Jun 1985-Operations Research
TL;DR: This paper uses signatures to describe a method for finding optimal assignments that terminates in at most n-1n-2/2 pivot steps and takes at most On3 work.
Abstract: The "signature" of a dual feasible basis of the assignment problem is an n-vector whose ith component is the number of nonbasic activities of type i, j. This paper uses signatures to describe a method for finding optimal assignments that terminates in at most n-1n-2/2 pivot steps and takes at most On3 work.

139 citations

"A competitive (dual) simplex method..." refers background or methods in this paper

  • ...The method is not, however, a simplex method in that a pivot can be made by deleting an edge (i,j) with x~ = 0. The method ignores the values of x(T), although it can be shown [ 1 ] that every pivot deletes an edge with x~j ~ /0, but the row signature is not (2, 2,..., 2, 1) and so the algorithm continues to pivot despite the fact that an optimal ......

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  • ...Since there are n - k- 1 row nodes of degree 2 at the beginning of level k, at most n - k pivots must produce a tree of level k - 1, and so at most (n - 1)(n -2)/2 pivots find an optimal basis T. Examples exist that show this is the best possible bound [ 1 ]....

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  • ...The first approach [ 1 ] uses only the signatures as guides to the choice of pivots from one dual feasible basis to a neighboring one....

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Related Papers (5)
The Hungarian method for the assignment problem

[...]

01 Mar 1955-Naval Research Logistics Quarterly
Harold W. Kuhn
The alternating basis algorithm for assignment problems

[...]

01 Dec 1977-Mathematical Programming
Richard S. Barr, Fred Glover, Darwin Klingman
A new algorithm for the assignment problem

[...]

01 Dec 1981-Mathematical Programming
Dimitri P. Bertsekas
Linear Programming and Extensions

[...]

01 Jan 1963
George B. Dantzig
A network simplex method

[...]

01 Dec 1976-Mathematical Programming
William H. Cunningham