Efficient dual simplex algorithms for the assignment problem

Efficient algorithms based upon Balinski's signature method are described for solving then × n assignment problem. These algorithms are special variants of the dual simplex method and are shown to have computational bounds of O(n ³). Variants for solving sparse assignment problems withm arcs that require O(m) space and O(mn + n ² logn) time in the worst case are also presented.This research was supported in part by the National Science Foundation under Grant No. MCS-8006064 and by the Army Research Office under Contracts No. DAAG 29-82-K0163 and DAAG 29-83-K0106     

A simple dual algorithm for the generalised assignment problem

A new algorithm for the generalised assignment problem is described in this paper. The dual-type algorithm uses a simple heuristic derived from a relaxation of the problem. The algorithm has been tested on generalised assignment problems of substantial size and compared to an exact integer programming approach and a well-established heuristic approach. Computational results look promising in terms of speed and solution quality.     

A competitive (dual) simplex method for the assignment problem

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).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 asx _ij < 0. A choice (i,j) over columnsj having at least 3 basic activities that minimizesx _ij is shown to converge in at most ( ₂ ^n-1 ) pivots, and at most O(n ³) time, and it is argued that on average the number of pivots is at mostn logn. Dedicated with affection to George B. Dantzig on the occasion of his seventieth birthday.     

A strongly polynomial simplex method for the linear fractional assignment problem

In this paper we show that the complexity of the simplex method for the linear fractional assignment problem (LFAP) is strongly polynomial. Although LFAP can be solved in polynomial time using various algorithms such as Newton’s method or binary search, no polynomial time bound for the simplex method for LFAP is known.     

A polynomial dual simplex algorithm for the generalized circulation problem

This paper presents a polynomial-time dual simplex algorithm for the generalized circulation problem. An efficient implementation of this algorithm is given that has a worst-case running time of O(m ²(m+nlogn)logB), where n is the number of nodes, m is the number of arcs and B is the largest integer used to represent the rational gain factors and integral capacities in the network. This running time is as fast as the running time of any combinatorial algorithm that has been proposed thus far for solving the generalized circulation problem. Received: June 1998 / Accepted: June 27, 2001?Published online September 17, 2001     

A simplex-based labelling algorithm for the linear fractional assignment problem

This paper constructs an algorithm to solve the fractional assignment problem. Algorithms that are currently used are mostly based on parametric approaches and must solve a sequence of optimization procedures. They also neglect the difficulties caused by degeneracy. The proposed algorithm performs optimization once and overcomes degeneracy. The main features of the algorithm are an effective initial heuristic approach, a simple labelling procedure and an implicit primal-dual schema. A numerical example is presented and demonstrates that the proposed algorithm is easy to apply. Computational results are compared with those from other developed methods. The results show that the proposed algorithm is efficient.     

On dual solutions of the linear assignment problem

For the linear assignment problem we describe how to obtain different dual solutions. It turns out that a shortest path algorithm can be used to compute such solutions with several interesting properties that enable to do better post-optimality analysis.Two examples illustrate how different dual solutions can be used in the context of the traveling salesman problem.     

A data parallel augmenting path algorithm for the dense linear many-to-one assignment problem

The purpose of this study is to describe a data parallel primal-dual augmenting path algorithm for the dense linear many-to-one assignment problem also known as semi-assignment. This problem could for instance be described as assigning n persons to m(n) job groups.The algorithm is tailored specifically for massive SIMD parallelism and employs, in this context, a new efficient breadth-first-search augmenting path technique which is found to be faster than the shortest augmenting path search normally used in sequential algorithms for this problem. We show that the best known sequential computational complexity of O(mn ² ) for dense problems, is reduced to the parallel complexity of O(mn), on a machine with n processors supporting reductions in O(1) time. The algorithm is easy to implement efficiently on commercially available massively parallel computers. A range of numerical experiments are performed on a Connection Machine CM200 and a MasPar MP-2. The tests show the good performance of the proposed algorithm.     

A new algorithm for the assignment problem

We propose a new algorithm for the classical assignment problem. The algorithm resembles in some ways the Hungarian method but differs substantially in other respects. The average computational complexity of an efficient implementation of the algorithm seems to be considerably better than the one of the Hungarian method. In a large number of randomly generated problems the algorithm has consistently outperformed an efficiently coded version of the Hungarian method by a broad margin. The factor of improvement increases with the problem dimensionN and reaches an order of magnitude forN equal to several hundreds.Work supported by Grant NSF ENG-7906332.     

A genetic algorithm for the generalised assignment problem

A new algorithm for the generalised assignment problem is described in this paper. The algorithm is adapted from a genetic algorithm which has been successfully used on set covering problems, but instead of genetically improving a set of feasible solutions it tries to genetically restore feasibility to a set of near-optimal ones. Thus it may be regarded as operating in a dual sense to the more familiar genetic approach. The algorithm has been tested on generalised assignment problems of substantial size and compared to an exact integer programming approach and a well-established heuristic approach.     

A dual algorithm for the one-machine scheduling problem

A branch and bound algorithm is presented for the problem of schedulingn jobs on a single machine to minimize tardiness. The algorithm uses a dual problem to obtain a good feasible solution and an extremely sharp lower bound on the optimal objective value. To derive the dual problem we regard the single machine as imposing a constraint for each time period. A dual variable is associated with each of these constraints and used to form a Lagrangian problem in which the dualized constraints appear in the objective function. A lower bound is obtained by solving the Lagrangian problem with fixed multiplier values. The major theoretical result of the paper is an algorithm which solves the Lagrangian problem in a number of steps proportional to the product ofn ² and the average job processing time. The search for multiplier values which maximize the lower bound leads to the formulation and optimization of the dual problem. The bounds obtained are so sharp that very little enumeration or computer time is required to solve even large problems. Computational experience with 20-, 30-, and 50-job problems is presented.     

A new greedy algorithm for the quadratic assignment problem

The classical greedy algorithm for discrete optimization problems where the optimal solution is a maximal independent subset of a finite ground set of weighted elements, can be defined in two ways which are dual to each other. The Greedy-In where a solution is constructed from the empty set by adding the next best element, one at a time, until we reach infeasibility, and the Greedy-Out where starting from the ground set we delete the next worst element, one at a time, until feasibility is reached. It is known that while the former provides an approximation ratio for maximization problems, its worst case performance is not bounded for minimization problems, and vice-versa for the later. However the Greedy-Out algorithm requires an oracle for checking the existence of a maximal independent subset which for most discrete optimization problems is a difficult task. In this work we present a Greedy-Out algorithm for the quadratic assignment problem by providing a combinatorial characterization of its solutions.     

A branch-and-bound algorithm for the singly constrained assignment problem

The singly constrained assignment problem (SCAP) is a linear assignment problem (LAP) with one extra side constraint, e.g., due to a time restriction. The SCAP is, in contrast to the LAP, difficult to solve. A branch-and-bound algorithm is presented to solve the SCAP to optimality. Lower bounds are obtained by Lagrangean relaxation. Computational results show that the algorithm is able to solve different types of SCAP instances up to size n = 1000 within short running times on a standard personal computer.     

A linear assignment approach for the least-squares protein morphing problem

This work addresses the computation of free-energy differences between protein conformations by using morphing (i.e., transformation) of a source conformation into a target conformation. To enhance the morphing procedure, we employ permutations of atoms: we seek to find the permutation σ that minimizes the mean-square distance traveled by the atoms. Instead of performing this combinatorial search in the space of permutations, we show that the best permutation can be found by solving a linear assignment problem. We demonstrate that the use of such optimal permutations significantly improves the efficiency of the free-energy computation.     

A new algorithm for geometric programming based on the linear structure of its dual problem

In this article, we present an algorithm for the resolution of a nonlinear optimization problem, concretely the posynomial geometric programming model. The solution procedure that we develop extends the condensation techniques for geometric programming, allowing us to find the optimal solutions to the dual geometric problems that we get from the interior of the corresponding feasible regions, in the line that interior point methods for linear programming work, which leads us to obtain considerable computational advantages with respect of the classical solution procedures.     

On optimality of a polynomial algorithm for random linear multidimensional assignment problem

We demonstrate that the linear multidimensional assignment problem with iid random costs is polynomially e{\varepsilon} -approximable almost surely (a.s.) via a simple greedy heuristic, for a broad range of probability distributions of the assignment costs. Specifically, conditions on discrete and continuous distributions of the cost coefficients, including distributions with unbounded support, have been established that guarantee convergence to unity in the a.s. sense of the cost ratio between the greedy solution and optimal solution. The corresponding convergence rates have been determined.     

Extensions of the sequential stochastic assignment problem

The sequential stochastic assignment problem (SSAP) allocates N workers to N IID sequentially arriving tasks so as to maximize the expected total reward. This paper studies two extensions of the SSAP. The first one assumes that the values of any two consecutive tasks are dependent on each other while the exact number of tasks to arrive is unknown until after the final arrival. The second extension generalizes the first one by assuming that the number of workers is also random. Optimal assignment policies for both problems are derived and proven to have the same threshold structure as the optimal policy of the SSAP.     

A rank-based approach to the sequential selection and assignment problem

In the classical sequential assignment problem, “machines” are to be allocated sequentially to “jobs” so as to maximize the expected total return, where the return from an allocation of job j to machine k is the product of the value x_j of the job and the weight p_k of the machine. The set of m machines and their weights are given ahead of time, but n jobs arrive in sequential order and their values are usually treated as independent, identically distributed random variables from a known univariate probability distribution with known parameter values. In the paper, we consider a rank-based version of the sequential selection and assignment problem that minimizes the sum of weighted ranks of jobs and machines. The so-called “secretary problem” is shown to be a special case of our sequential assignment problem (i.e., m = 1). Due to its distribution-free property, our rank-based assignment strategy can be successfully applied to various managerial decision problems such as machine scheduling, job interview, kidney allocations for transplant, and emergency evacuation plan of patients in a mass-casualty situation.