Classical Assignment Problem Research Articles

Partition-distance via the assignment problem

Accuracy testing of various pedigree reconstruction methods requires an efficient algorithm for the calculation of distance between a known partition and its reconstruction. The currently used algorithm of Almudevar and Field takes a prohibitively long time for certain partitions and population sizes. We present an algorithm that very efficiently reduces the partition-distance calculation to the classic assignment problem of weighted bipartite graphs that has known polynomial-time solutions. The performance of the algorithm is tested against the Almudevar and Field partition-distance algorithm to verify the significant improvement in speed. Computer code written in java is available upon request from the first author.

Using assignment model as an automated recovery system

PurposeThis paper aims to deal with disaster and recovery systems, introducing the catastrophe matrix as a disaster‐preventing tool.Design/methodology/approachA new model is proposed which is proved to be a classical assignment problem. It is solved based on well‐known optimization methods. Research is presented for the danger risk of the system and a fully automated recovery system with a minimum cost is presented.FindingsResearch accepts the fact that a catastrophe has already been activated on the system.Research limitations/implicationsThe difficulty of solving the proposed model of the threat and countermeasures still remains.Originality/valueThe proposed models show a new original way to confront dangers and overcome the existing classical models based on risk management.

Etude on theme of Dijkstra

We present a tutorial for undergraduate students who are interested in different programming contests like National Programming Olympiads or regional ACM International Collegiate Programming Contests (ACM ICPC). It is based on a problem discussed by E. W. Dijkstra in a public lecture in 1994. The tutorial was a part of a special undergraduate course presented for students of Novosibirsk State University in 2003-04 academic year during their preparation for ACM ICPC Final at Prague (2004). The original problem is how to couple <i>N</i> black and <i>N</i> white points on the plane by intervals without intersections. E. W. Dijkstra suggested a nice heuristic search algorithm and proved its termination and correctness by the method of R. W. Floyd. Unfortunately, the upper bound of his algorithm was exponential. Due to this reason, he suggested to audience investigation of the problem complexity and development of an efficient algorithm. We demonstrate that the problem is decidable in cubic time by quadratic reduction to the classical assignment problem. We also discuss how to solve the coupling problem in pure geometric/topological way. We also present some examples of utility of the coupling problem.

Colony location algorithm for assignment problems

A novel algorithm called Colony Location Algorithm (CLA) is proposed. It mimics the phenomena in biotic community that colonies of species could be located in the places most suitable to their growth. The factors working on the species location such as the nutrient of soil, resource competition between species, growth and decline process, and effect on environment were considered in CLA via the nutrient function, growth and decline rates, environment evaluation and fertilization strategy. CLA was applied to solve the classical assignment problems. The computation results show that CLA can achieve the optimal solution with higher possibility and shorter running time.

An Improved Version of a Multiple Target Tracking Algorithm

NEWNAM 6 and with the more recent work of MILLER, STONE and COX. 7 In the last of these works, following the Murty‘s method the authors implement three optimizations, producing a speedup by factor of over 20. On the other hand, the measurements-to-target association as part of a frame of MHT approach can be successfully formulated as a classical assignment problem. So, using an algorithm capable to find exact first K-best solutions of the so formulated assignment problem gives us K hypotheses of highest probability without first generating all feasible hypotheses and then pruning them. 3

Traffic assignment: Equilibrium models

The formulation of static traffic assignment models, with variable and fixed demand, based on Wardrop's first principle, are presented for deterministic and stochastic models. The main algorithms used to obtain solutions for these network equilibrium models are given for each class of assignment problems. Calibration and validation issues are considered.

Efficient algorithms and codes for k-cardinality assignment problems

Given a cost matrix W and a positive integer k, the k-cardinality assignment problem is to assign k rows to k columns so that the sum of the corresponding costs is a minimum. This generalization of the classical assignment problem is solvable in polynomial time, either by transformation to min-cost flow or through specific algorithms. We consider the algorithm recently proposed by Dell'Amico and Martello for the case where W is dense, and derive, for the case of sparse matrices, an efficient algorithm which includes original heuristic preprocessing techniques. The resulting code is experimentally compared with min-cost flow codes from the literature. Extensive computational tests show that the code is considerably faster, and effectively solves very large sparse and dense instances.

On the variable-depth-search heuristic for the linear-cost generalized assignment problem *

The classical assignment problem seeks to determine a mapping between a set of assignees and a set of assignments. The linear cost assignment problem (LCGAP), as a generalized model, incorporates the relative workloads between assignees and assignments. Although LCGAP is computationally intractable, it has been extensively studied because of its practical implications in many real world situations. Variable-depth-search heuristic (VDSH) method is one of the solution methods that have been developed to produce quality near-optimal solutions to LCGAP. The main structure of VDSH consists of two basic operations: reassign and swap. In this paper, we make further observations on this effective heuristic method through a series of computational experiments. The numerical results statistically evince that different combina-tions of these two operations will result in solutions of different quality levels. These findings are expected to have similar implications to search heuristics for other optimization problems.

Perturbation-minimizing frequency assignment in a changing TDMA/FDMA cellular environment

A robust operation of assigning frequencies to requirements in a time-division multiple-access (TDMA) [or frequency-division multiple-access (FDMA)] cellular system should cope with environmental changes such as short-term demand rises and long-term capacity expansions while always keeping the required realignment process as simple as possible. In this paper, we consider the so-called perturbation-minimizing frequency assignment problem (PMFAP), the objective of which is to assign available frequencies for newly generated requirements with the minimum change in the existing frequency assignments while meeting the interference-related constraints. For PMFAP, we propose a heuristic algorithm based on the so-called Bv-Dw rearrangement technique, which can also be applied to the classic frequency assignment problem (PAP) with a slight modification. Two kinds of computational experiments, one for each of the above two problems, are performed to demonstrate the powerful features of the proposed solution method not only in its suitability for real-world frequency management, but also in solving the FAP.

Performance of the MOSA Method for the Bicriteria Assignment Problem

The classical linear Assignment problem is considered with two objectives. The aim is to generate the set of efficient solutions. An exact method is first developed based on the two-phase approach. In the second phase a new upper bound is proposed so that larger instances can be solved exactly. The so-called MOSA (Multi-Objective Simulated Annealing) is then recalleds its efficiency is improved by initialization with a greedy approach. Its results are compared to those obtained with the exact method. Extensive numerical experiments have been realized to measure the performance of the MOSA method.

The Assignment Problem with Seniority and Job Priority Constraints

Consider an assignment problem in which persons are qualified for some but usually not all of the jobs. Moreover, assume persons belong to given seniority classes and jobs have given priority levels. Seniority constraints impose that the solution be such that no unassigned person can be given a job unless an assigned person with the same or higher seniority becomes unassigned. Priority constraints specify that the solution must be such that no unassigned job can become assigned without a job with the same or higher priority becoming unassigned. It is shown that: (i) adding such constraints does not reduce and may even increase the number of assigned persons in the optimal solution; (ii) using a greedy heuristic for constrained assignment (as often done in practice) may reduce the number of assigned persons by half, and (iii) an optimal solution to the assignment problem with both types of constraints can be obtained by solving a classical assignment problem with adequately modified coefficients.

Solution of the Cumulative Assignment Problem With a Well-Structured TabuSearch Method

The Cumulative Assignment Problem is an NP-complete problem obtained by substituting the linear objective function of the classic Linear Assignment Problem, with a non-linear cumulative function. In this paper we present a first attempt to solve the Cumulative Assignment Problem with metaheuristic techniques. In particular we consider two standard techniques, namely the Simulated Annealing and the Multi-Start methods, and we describe the eXploring Tabu Search: a new structured Tabu Search algorithm which uses an iterative multi-level approach to improve the search. The new method is analyzed through extensive computational experiments and proves to be more effective than the standard methods.

On a Class of Assignment Problems

In the Generalized Assignment Problem, It requires to minimize the cost of assignment of ‘n’ tasks to a subset of ‘m’ agents. Each task is to be assigned to one agent and each agent is limited only a maximum amount of resources available to him. A Lexi Search algorithm using Pattern Recognition was developed and presented already by the authors, along with computational experience. In the present paper, fresh computational experience is reported along with the refinements made in the algortihm.

The assignment problem with external interactions

The classical assignment problem matches n jobs to n machines in a way that minimizes total assignment costs. To allow for the possibility of diverting internal jobs outside the machine shop and accepting external jobs into the machine shop, we define the assignment problem with external interactions (APEX) as a linear assignment problem with a single unrestricted arc. A feasible solution of APEX may involve up to twice as many assignments as an assignment problem of the same size. An efficient sequential shortest path algorithm is developed for APEX. The algorithm uses initialization heuristics to obtain a dual feasible solution that satisfies the complementary slackness conditions with a partial set of assignments. Primal feasibility is then obtained through a shortest augmenting path algorithm that makes the remaining assignments. As APEX can be formulated as a minimum-cost network flow problem and as a standard assignment problem of larger size, we compare the proposed algorithm with state-of-the-art network flow and assignment approaches. The algorithm developed here has much smaller storage requirements than have alternative approaches. In addition, extensive computational results demonstrate that it is from 2 to 52 times faster.

A comparison of two algorithms for determining ranked assignments with application to multitarget tracking and motion correspondence

Recently, it has become clear that determining a ranked set of assignments allows computation of very good approximations to the data association problem. Several algorithms have been proposed but only two return the k-best assignments in reasonable time. One is Danchick and Newnams' [1993] algorithm, which is based on the recognition that determining the best assignment is a classical assignment problem and that determining a ranked set of assignments may be accomplished by solving a series of modified copies of the initial assignment problem. The other algorithm is originally due to Murty [1968] and was most recently described within the context of multitarget tracking. We evaluate the two algorithm using randomly generated data and data obtained from an electrooptical sensor simulation in which 90 missiles are launched. These evaluations show that Murty's algorithm perform significantly better in all scenarios. We show the relationship between the two algorithms and how Danchick and Newnam's algorithm can be very easily modified to Murty's algorithm. Experimental results using Murty's algorithm suggest that a solution to the real-time data association problem is now feasible.

The Use of Specially Structured Models for Obtaining Bounds in the Quadratic Assignment Problem

In this paper we study a procedure for finding bounds for the quadratic assignment problem. This procedure may be used as a sub-routine in hybrid procedures for solving this problem. The approach is based upon a data decomposition method, linking the actual data to the data of a special class of assignment problems for which bounds are computationally tractable.

Parallel Asynchronous Hungarian Methods for the Assignment Problem

In this paper, we discuss the parallel asynchronous implementation of the Hungarian method for solving the classical assignment problem. Multiple augmentations and price rises are simultaneously attempted starting from several unassigned sources and using possibly outdated price and assignment information. The results are then merged asynchronously subject to rather weak compatibility conditions. We show the validity of this algorithm and we demonstrate computationally that an asynchronous implementation is often faster than its synchronous counterpart. INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

Deterministic multiprocessor scheduling with multiple objectives

One common job scheduling objective is to minimize makespan. The problem can be modeled as integer linear programming (ILP) and will often have multiple alternative optimal solutions. However, secondary considerations, e.g. minimizing the second latest completion time, may very well dictate a preference between solutions with identical makespans. Because of the nature of the primary objective, standard approaches for optimizing a prioritized set of multiple objectives will not work. In this paper we prove that for unit time problems, an appropriate objective function can be formulated, which, when optimized, satisfies both the primary and secondary objectives. Moreover, the new formulation can be modeled as a classical assignment problem (AP). This has the added advantage of efficiency of solution and availability of software. Applications to computer processor scheduling and the military are presented.

On a new algorithm for stable assignment*

The concept of stable assignment comes from a classical assignment problem called the Stable Marriage Assignment. Suppose that there are two disjoint sets, one consists of men and the other of women. Each member of a set ranks the members of the other set in order of preference. Matching persons between two sets is a one-to-one relation. If there does not exist a random pair both preferring each other to their assigned partner, the assignment is said to be stable. In a recent paper. Huang and Chan generalized the above one-to-one type assignment problem to a multi-function assignment problem and proposed an algorithm for finding stable assignments for both problems. In this paper, we present a couterexample for which the assignment obtained by that algorithm is not stable.

Parallel synchronous and asynchronous implementations of the auction algorithm

In this paper we discuss the parallel implementation of the auction algorithm for the classical assignment problem. We show that the algorithm admits a totally asynchronous implementation and we consider several implementations on a shared memory machine, with varying degrees of synchronization. We also discuss and explore computationally the tradeoffs involved in using asynchronism to reduce the synchronization penalty.