Assignment problem

About: Assignment problem is a research topic. Over the lifetime, 7588 publications have been published within this topic receiving 172820 citations. The topic is also known as: marriage problem.

Fast approximate quadratic programming for graph matching.

Abstract: Quadratic assignment problems arise in a wide variety of domains, spanning operations research, graph theory, computer vision, and neuroscience, to name a few. The graph matching problem is a special case of the quadratic assignment problem, and graph matching is increasingly important as graph-valued data is becoming more prominent. With the aim of efficiently and accurately matching the large graphs common in big data, we present our graph matching algorithm, the Fast Approximate Quadratic assignment algorithm. We empirically demonstrate that our algorithm is faster and achieves a lower objective value on over 80% of the QAPLIB benchmark library, compared with the previous state-of-the-art. Applying our algorithm to our motivating example, matching C. elegans connectomes (brain-graphs), we find that it efficiently achieves performance.

A Lagrangian Relaxation Algorithm for Multidimensional Assignment Problems Arising from Multitarget Tracking

Abstract: The central problem in multitarget tracking is the data association problem of partitioning the observations into tracks in some optimal way so that an accurate estimate of the true tracks can be recovered. This work considers what is perhaps the simplest multitarget tracking problem in a setting where the issues are easily delineated, i.e., straight lines in two-dimensional space-time with an error component introduced into the observations. A multidimensional assignment problem is formulated using gating techniques to introduce sparsity into the problem and filtering techniques to generate tracks which are then used to score each assignment of a collection of observations to a filtered track. Problem complexity is further reduced by decomposing the problem into disjoint components, which can then be solved independently. A recursive Lagrangian relaxation algorithm is developed to obtain high quality suboptimal solutions in real-time. The algorithms are, however, applicable to a large class of sparse mul...

A restricted Lagrangean approach to the traveling salesman problem

Abstract: We describe an algorithm for the asymmetric traveling salesman problem (TSP) using a new, restricted Lagrangean relaxation based on the assignment problem (AP). The Lagrange multipliers are constrained so as to guarantee the continued optimality of the initial AP solution, thus eliminating the need for repeatedly solving AP in the process of computing multipliers. We give several polynomially bounded procedures for generating valid inequalities and taking them into the Lagrangean function with a positive multiplier without violating the constraints, so as to strengthen the current lower bound. Upper bounds are generated by a fast tour-building heuristic. When the bound-strengthening techniques are exhausted without matching the upper with the lower bound, we branch by using two different rules, according to the situation: the usual subtour breaking disjunction, and a new disjunction based on conditional bounds. We discuss computational experience on 120 randomly generated asymmetric TSP's with up to 325 cities, the maximum time used for any single problem being 82 seconds. This is a considerable improvement upon earlier methods. Though the algorithm discussed here is for the asymmetric TSP, the approach can be adapted to the symmetric TSP by using the 2-matching problem instead of AP.

The Dynamic Assignment Problem

Abstract: There has been considerable recent interest in the dynamic vehicle routing problem, but the complexities of this problem class have generally restricted research to myopic models. In this paper, we address the simpler dynamic assignment problem, where a resource (container, vehicle, or driver) can serve only one task at a time. We propose a very general class of dynamic assignment models, and propose an adaptive, nonmyopic algorithm that involves iteratively solving sequences of assignment problems no larger than what would be required of a myopic model. We consider problems where the attribute space of future resources and tasks is small enough to be enumerated, and propose a hierarchical aggregation strategy for problems where the attribute spaces are too large to be enumerated. Finally, we use the formulation to also test the value of advance information, which offers a more realistic estimate over studies that use purely myopic models.

The alternating basis algorithm 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.