Signature Methods for the Assignment Problem

Balinski, Michel

doi:10.1287/opre.33.3.527

Cited by 120 publications

(80 citation statements)

References 11 publications

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“…Many of these algorithms make use of so-called signatures, introduced by Balinski [18]. Therefore, such algorithms are also called signature methods.…”

Section: Dual Simplex-based Algorithmsmentioning

confidence: 99%

“…, 2). Balinski [18,19] proposes two algorithms to find such a tree T . Both algorithms start with a tree with row signature (n, 1, .…”

Section: Definition 34mentioning

confidence: 99%

“…, 1) and dual variables u 1 = 0, v j = c 1j for 1 ≤ j ≤ n, and u i = min j {c ij − v j }, for 2 ≤ i ≤ n. Such a tree is often called a Balinski tree. In the first algorithm [18] the choice of the pivoting edges e and f is guided only by the signatures, trying to decrease the number of 1-s in the signature. This makes the algorithm a nongenuine simplex method because it may pivot on some edge e = (k, l) with x kl = 0.…”

Section: Definition 34mentioning

confidence: 99%

“…Clearly, this algorithm is not a genuine simplex dual algorithm for the same reasons as Balinski's algorithm. The second algorithm of Balinski [19] employs another pivot rule which is a combination of the rule based on signatures used in [18] and the classical pivot on an edge e = (i, j) with the most negative x ij . This rule allows the algorithm to visit only so-called strongly dual feasible trees.…”

Section: Definition 34mentioning

confidence: 99%

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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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“…Many of these algorithms make use of so-called signatures, introduced by Balinski [18]. Therefore, such algorithms are also called signature methods.…”

Section: Dual Simplex-based Algorithmsmentioning

confidence: 99%

“…, 2). Balinski [18,19] proposes two algorithms to find such a tree T . Both algorithms start with a tree with row signature (n, 1, .…”

Section: Definition 34mentioning

confidence: 99%

Section: Definition 34mentioning

confidence: 99%

Section: Definition 34mentioning

confidence: 99%

See 2 more Smart Citations

Linear Assignment Problems and Extensions

Burkard

Çela

1999

Handbook of Combinatorial Optimization

255

150

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“…Due to its theoretical and practical significance, various techniques have been proposed for obtaining an optimal solution. Most rely on iterative improvement of some cost function [17]- [20], while Kuhn's Hungarian algorithm [21] was the first method specifically designed for that problem. All above techniques require a single central processor that handles all computation and information in the system.…”

Section: Introductionmentioning

confidence: 99%

A distributed auction algorithm for the assignment problem

Zavlanos

Spesivtsev

Pappas

2008

2008 47th IEEE Conference on Decision and Control

175

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Abstract-The assignment problem constitutes one of the fundamental problems in the context of linear programming. Besides its theoretical significance, its frequent appearance in the areas of distributed control and facility allocation, where the problems' size and the cost for global computation and information can be highly prohibitive, gives rise to the need for local solutions that dynamically assign distinct agents to distinct tasks, while maximizing the total assignment benefit. In this paper, we consider the linear assignment problem in the context of networked systems, where the main challenge is dealing with the lack of global information due to the limited communication capabilities of the agents. We address this challenge by means of a distributed auction algorithm, where the agents are able to bid for the task to which they wish to be assigned. The desired assignment relies on an appropriate selection of bids that determine the prices of the tasks and render them more or less attractive for the agents to bid for. Up to date pricing information, necessary for accurate bidding, can be obtained in a multi-hop fashion by means of local communication between adjacent agents. Our algorithm is an extension to the parallel auction algorithm proposed by Bertsekas et al to the case where only local information is available and it is shown to always converge to an assignment that maximizes the total assignment benefit within a linear approximation of the optimal one.

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Order-preserving assignments

Padberg

Alevras

1994

Naval Research Logistics

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Given an ordered list of n items, p possible positions, and integer profits cij for assigning item i to position j, the order‐preserving assignment problem consists of finding a profit‐optimal assignment that assigns items to contiguous positions while preserving the original ranking, i.e., the highest‐ranked item that will be assigned is assigned to position 1 and so on. Positions must be assigned contiguously starting with position one, but items need not be assigned to any particular position. We formulate the problem as a zero‐one linear program, derive a minimal description of the associated polytope by linear inequalities, show that the diameter of the polytope equals two, and give a linear‐time algorithm for the optimization problem, i.e., one that runs in time that is linear in the number of variables of the problem. We do this for both cases where at most p positions and where exactly p positions must be assigned and discuss briefly two modifications of the basic model. © 1994 John Wiley & Sons, Inc.

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Signature Methods for the Assignment Problem

Cited by 120 publications

References 11 publications

Linear Assignment Problems and Extensions

Linear Assignment Problems and Extensions

A distributed auction algorithm for the assignment problem

Order-preserving assignments

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