Showing papers on "Assignment problem published in 2001"

Continuous-time analysis, eigenstructure assignment, and H/sub 2/ synthesis with enhanced linear matrix inequalities (LMI) characterizations

Abstract: This note describes a new framework for the analysis and synthesis of control systems, which constitutes a genuine continuous-time extension of results that are only available in discrete time. In contrast to earlier results the proposed methods involve a specific transformation on the Lyapunov variables and a reciprocal variant of the projection lemma, in addition to the classical linearizing transformations on the controller data. For a wide range of problems including robust analysis and synthesis, multichannel H/sub 2/ stateand output-feedback syntheses, the approach leads to potentially less conservative linear matrix inequality (LMI) characterizations. This comes from the fact that the technical restriction of using a single Lyapunov function is to some extent ruled out in this new approach. Moreover, the approach offers new potentials for problems that cannot be handled using earlier techniques. An important instance is the eigenstructure assignment problem blended with Lyapunov-type constraints which is given a simple and tractable formulation.

Bipartite graph partitioning and data clustering

Abstract: Many data types arising from data mining applications can be modeled as bipartite graphs, examples include terms and documents in a text corpus, customers and purchasing items in market basket analysis and reviewers and movies in a movie recommender system. In this paper, we propose a new data clustering method based on partitioning the underlying bipartite graph. The partition is constructed by minimizing a normalized sum of edge weights between unmatched pairs of vertices of the bipartite graph. We show that an approximate solution to the minimization problem can be obtained by computing a partial singular value decomposition (SVD) of the associated edge weight matrix of the bipartite graph. We point out the connection of our clustering algorithm to correspondence analysis used in multivariate analysis. We also briefly discuss the issue of assigning data objects to multiple clusters. In the experimental results, we apply our clustering algorithm to the problem of document clustering to illustrate its effectiveness and efficiency.

The ζ (2) limit in the random assignment problem

Abstract: Author(s): Aldous, DJ | Abstract: The random assignment (or bipartite matching) problem asks about An = minπ ∑ni=1 c(i, π(i)) where (c(i, j)) is a n × n matrix with i.i.d. entries, say with exponential(1) distribution, and the minimum is over permutations π. Mezard and Parisi (1987) used the replica method from statistical physics to argue nonrigorously that EAn → ζ(2) = π2/6. Aldous (1992) identified the limit in terms of a matching problem on a limit infinite tree. Here we construct the optimal matching on the infinite tree. This yields a rigorous proof of the ζ(2) limit and of the conjectured limit distribution of edge-costs and their rank-orders in the optimal matching. It also yields the asymptotic essential uniqueness property: every almost-optimal matching coincides with the optimal matching except on a small proportion of edges. © 2001 John Wiley a Sons, Inc. Random Struct. Alg., 18, 381-418, 2001.

Geometry helps in bottleneck matching and related problems

Abstract: Let A and B be two sets of n objects in \reals d , and let Match be a (one-to-one) matching between A and B . Let min(Match ), max(Match ), and Σ(Match) denote the length of the shortest edge, the length of the longest edge, and the sum of the lengths of the edges of Match , respectively. Bottleneck matching— a matching that minimizes max(Match )— is suggested as a convenient way for measuring the resemblance between A and B . Several algorithms for computing, as well as approximating, this resemblance are proposed. The running time of all the algorithms involving planar objects is roughly O(n 1.5 ) . For instance, if the objects are points in the plane, the running time of the exact algorithm is O(n 1.5 log n ) . A semidynamic data structure for answering containment problems for a set of congruent disks in the plane is developed. This data structure may be of independent interest. Next, the problem of finding a translation of B that maximizes the resemblance to A under the bottleneck matching criterion is considered. When A and B are point-sets in the plane, an O(n 5 log n) -time algorithm for determining whether for some translated copy the resemblance gets below a given ρ is presented, thus improving the previous result of Alt, Mehlhorn, Wagener, and Welzl by a factor of almost n . This result is used to compute the smallest such ρ in time O(n 5 log 2 n ) , and an efficient approximation scheme for this problem is also given. The uniform matching problem (also called the balanced assignment problem, or the fair matching problem) is to find Match * U , a matching that minimizes max (Match)-min(Match) . A minimum deviation matching Match * D is a matching that minimizes (1/n)Σ(Match) - min(Match) . Algorithms for computing Match * U and Match * D in roughly O(n 10/3 ) time are presented. These algorithms are more efficient than the previous O(n 4 ) -time algorithms of Martello, Pulleyblank, Toth, and de Werra, and of Gupta and Punnen, who studied these problems for general bipartite graphs.

m-best S-D assignment algorithm with application to multitarget tracking

Abstract: In this paper we describe a novel data association algorithm, termed m-best S-D, that determines in O(mSkn/sup 3/) time (m assignments, S/spl ges/3 lists of size n, k relaxations) the (approximately) m-best solutions to an S-D assignment problem. The m-best S-D algorithm is applicable to tracking problems where either the sensors are synchronized or the sensors and/or the targets are very slow moving. The significance of this work is that the m-best S-D assignment algorithm (in a sliding window mode) can provide for an efficient implementation of a suboptimal multiple hypothesis tracking (MHT) algorithm by obviating the need for a brute force enumeration of an exponential number of joint hypotheses. We first describe the general problem for which the m-best S-D applies. Specifically, given line of sight (LOS) (i.e., incomplete position) measurements from S sensors, sets of complete position measurements are extracted, namely, the 1st, 2nd, ..., mth best (in terms of likelihood) sets of composite measurements are determined by solving a static S-D assignment problem. Utilizing the joint likelihood functions used to determine the m best S-D assignment solutions, the composite measurements are then quantified with a probability of being correct using a JPDA-like (joint probabilistic data association) technique. Lists of composite measurements from successive scans, along with their corresponding probabilities, are used in turn with a state estimator in a dynamic 2-D assignment algorithm to estimate the states of moving targets over time. The dynamic assignment cost coefficients are based on a likelihood function that incorporates the "true" composite measurement probabilities obtained from the (static) m-best S-D assignment solutions. We demonstrate the merits of the m-best S-D algorithm by applying it to a simulated multitarget passive sensor track formation and maintenance problem, consisting of multiple time samples of LOS measurements originating from multiple (S=7) synchronized high frequency direction finding sensors.

A bi-level programming approach for trip matrix estimation and traffic control problems with stochastic user equilibrium link flows

Abstract: This paper deals with two mathematically similar problems in transport network analysis: trip matrix estimation and traffic signal optimisation on congested road networks. These two problems are formulated as bi-level programming problems with stochastic user equilibrium assignment as the second-level programming problem. We differentiate two types of solutions in the combined matrix estimation and stochastic user equilibrium assignment problem (or the combined signal optimisation and stochastic user equilibrium assignment problem): one is the solution to the bi-level programming problem and the other the mutually consistent solution where the two sub-problems in the combined problem are solved simultaneously. In this paper, we shall concentrate on the bi-level programming approach, although we shall also consider mutually consistent solutions so as to contrast the two types of solutions. The purpose of the paper is to present a solution algorithm for the two bi-level programming problems and to test the algorithm on several networks.

A probabilistic analysis for the range assignment problem in ad hoc networks

Abstract: In this paper we consider the following problem for ad hoc networks: assume that n nodes are distributed in a d-dimensional region, with 1≤d≤3, and assume that all the nodes have the same transmitting range r; how large must r be to ensure that the resulting network is strongly connected? We study this problem by means of a probabilistic approach, and we establish lower and upper bounds on the probability of connectedness. For the one-dimensional case, these bounds allow us to determine a suitable magnitude of r for a given number of nodes and displacement region size. In an alternate formulation, the bounds allow us to calculate how many nodes must be distributed should the transmitting range be fixed. Finally, we investigate the required magnitude of r in the two- and three-dimensional cases through simulation. Based on the bounds provided and on the simulation analysis, we conclude that, as compared to the deterministic case, a probabilistic solution to this range assignment problem achieves substantial energy savings. A number of other potential uses for our analyses are discussed as well

The airport gate assignment problem: mathematical model and a tabu search algorithm

Abstract: Considers an airport gate assignment problem that dynamically assigns airport gates to scheduled flights based on passengers' daily origin and destination flow data. The objective of the problem is to minimize the overall connection times during which passengers walk to catch their connection flights. We formulate this problem as a mixed 0-1 quadratic integer programming problem and then reformulate it as a mixed 0-1 integer problem with a linear objective function and constraints. We design a simple tabu search meta-heuristic to solve the problem. The algorithm exploits the special properties of different types of neighborhood moves, and create highly effective candidate list strategies. We also address issues of tabu short-term memory, dynamic tabu tenure, aspiration rules, and various intensification and diversification strategies. Preliminary computational experiments are conducted, and the results are presented and analyzed.

A Modeling Framework for Passenger Assignment on a Transport Network with Timetables

Abstract: This paper presents a new graph theoretic framework for the passenger assignment problem that encompasses simultaneously the departure time and the route choice. The implicit FIFO access to transit lines is taken into account by the concept of available capacity. This notion of flow priority has not been considered explicitly in previous models. A traffic equilibrium model is described and a computational procedure based on asymmetric boarding penalty functions is suggested.

A Note on Optimal Matching With Variable Controls Using the Assignment Algorithm

Abstract: The assignment algorithm is an old, well-known, widely implemented, fast, combinatorial algorithm for optimal matching in a bipartite graph. This note proposes a method for using the assignment algorithm to solve the problem of optimal matching with a variable number of controls, in which there is a choice not only of who to select as a control for each treated subject, but also of how many controls to have for each treated subject. The strategy uses multiple copies of treated subjects and sinks with zero cost to absorb extra controls. Also, it is shown that an optimal matching with variable numbers of controls cannot be obtained by starting with an optimal pair matching and adding the closest additional controls. An example involving mortality after surgery in Pennsylvania hospitals is used to illustrate the method.

A Set of Greedy Randomized Adaptive Local Search Procedure (GRASP) Implementations for the Multidimensional Assignment Problem

Abstract: The focal problem for centralized multisensor multitarget tracking is the data association problem of partitioning the observations into tracks and false alarms so that an accurate estimate of the true tracks can be recovered. Large classes of these association problems can be formulated as multidimensional assignment problems, which are known to be NP-hard for three dimensions or more. The assignment problems that result from tracking are large scale, sparse and noisy. Solution methods must execute in real-time. The Greedy Randomized Adaptive Local Search Procedure (GRASP) has proven highly effective for solving many classes NP-hard optimization problems. This paper introduces four GRASP implementations for the multidimensional assignment problem, which are combinations of two constructive methods (randomized reduced cost greedy and randomized max regret) and two local search methods (two-assignment-exchange and variable depth exchange). Numerical results are shown for a two random problem classes and one tracking problem class.

Models and a genetic algorithm for static aircraft-gate assignment problem

Abstract: Assigning aircraft to available gates at an airport can have a major impact on the efficiency of flight schedules and on the level of passenger satisfaction with the service. Unexpected changes, due to air traffic delays, severe weather conditions, or equipment failures, may disrupt the initial assignments and compound the difficulty of maintaining smooth station operations. Recently, mathematical models and procedures (optimal and heuristic) have been proposed to provide solutions with minimum dispersion of idle time periods for static aircraft-gate assignment problems. This paper introduces a unified framework to specifically treat the objective functions of the previous models. It also provides linear representations of these models and identifies the conditions under which the optimal solutions can be obtained in polynomial time. Furthermore, a genetic algorithm utilizing problem specific knowledge is proposed to provide effective alternative solutions.

On Bipartite Drawings and the Linear Arrangement Problem

Abstract: The bipartite crossing number problem is studied and a connection between this problem and the linear arrangement problem is established A lower bound and an upper bound for the optimal number of crossings are derived, where the main terms are the optimal arrangement values Two polynomial time approximation algorithms for the bipartite crossing number are obtained The performance guarantees are O(log n) and O(log2 n) times the optimal, respectively, for a large class of bipartite graphs on n vertices No polynomial time approximation algorithm which could generate a provably good solution had been known For a tree, a formula is derived that expresses the optimal number of crossings in terms of the optimal value of the linear arrangement and the degrees, resulting in an O(n16) time algorithm for computing the bipartite crossing number The problem of computing a maximum weight biplanar subgraph of an acyclic graph is also studied and a linear time algorithm for solving it is derived No polynomial time algorithm for this problem was known, and the unweighted version of the problem had been known to be NP-hard, even for planar bipartite graphs of degree at most 3

Solving quadratic assignment problems using convex quadratic programming relaxations

Abstract: We describe a branch-and-bound algorithm for the quadratic assignment problem (QAP) that uses a convex quadratic programming (QP) relaxation to obtain a bound at each node. The QP subproblems are approximately solved using the Frank-Wolfe algorithm, which in this case requires the solution of a linear assignment problem on each iteration. Our branching strategy makes extensive use of dual information associated with the QP subproblems. We obtain state-of-the-art computational results on large benchmark QAPs

On the wavelength assignment problem in multifiber WDM star and ring networks

Abstract: This paper studies-the off-line wavelength assignment problem in star and ring networks that deploy multiple fibers between nodes and use wavelength division multiplexing (WDM) for transmission. The results in this paper show that the ability to switch between fibers increases wavelength utilization. In particular, sharper per-fiber bounds on the number of required wavelengths are derived for the multifiber version of the assignment problem in star and ring networks. Additionally, the complexity of the problem is studied and several constrained versions of the problem are also considered for star and ring networks. A summary of contributions is provided.

A Case-Based Reasoning Approach for Due-Date Assignment in a Wafer Fabrication Factory

Abstract: This study explores a new application of Case-Based Reasoning (CBR) in the due-date assignment problem of the wafer fabrication factory. Owing to the complexity of the wafer fabrication, the manufacturing processes of the wafer are very complicated and time-consuming. Thus, the due-date assignment of each order presents a challenging problem to the production planning and scheduling people. Since the product of each order is closely related to the products manufactured before, the CBR approach provides a good tool for us to apply it to the due-date assignment problem. The CBR system could potentially replace the human decision in the estimation of the due-date. Therefore, a CBR system is developed in this study using the similarity coefficient of each order with previous orders. The experimental results show that the proposed approach is very effective and comparable with a neural network approach.

Combinatorial approximation algorithms for the maximum directed cut problem

Abstract: We describe several combinatorial algorithms for the maximum directed cut problem. Among our results is a simple linear time 9/20-approximation algorithm for the problem, and a somewhat slower ½-approximation algorithm that uses a bipartite matching routine. No better combinatorial approximation algorithms are known even for the easier maximum cut problem for undirected graphs. Our algorithms do not use linear programming, nor semidefinite programming. They are based on the observation that the maximum directed cut problem is equivalent to the problem of finding a maximum independent set in the line graph of the input graph, and that the linear programming relaxation of the problem is equivalent to the problem of finding a maximum fractional independent set of that line graph. The maximum fractional independent set problem can be easily reduced to a bipartite matching problem. As a consequence of this relation, we also get that the maximum directed cut problem for bipartite digraphs can be solved in polynomial time.

Analytical models of the dynamic traffic assignment problem

Abstract: A review of analytical formulations of the dynamic traffic assignment problem is presented, focusing on the authors' experience with variational inequality approaches. Solution algorithms and computational issues requiring additional study are discussed.

A path-based traffic assignment algorithm based on the TRANSYT traffic model

Abstract: This paper presents a path-based traffic assignment formulation and its solution algorithm for solving an asymmetric traffic assignment problem based on the TRANSYT traffic model, a well-known procedure to determine the queues and delays in a signal-controlled network with explicit considerations of the signal co-ordination effects and platoon dispersion on the streets. The solution algorithm employs a Frank–Wolfe method to identify the descent direction at each iteration, which requires the input of the derivatives information. A post-simulation sensitivity analysis is developed to estimate the derivatives in the TRANSYT traffic model. Good agreement of results with the values determined by numerical differentiation is obtained. Using these derivatives information, the Frank–Wolfe method shows a good convergence behavior to the equilibrium solution. Comparison with other methods is also discussed in a numerical example to demonstrate the effectiveness of the proposed methodology.

Address code generation for digital signal processors

Abstract: In this paper we propose a procedure to generate code with minimum number of addressing instructions. We analyze different methods of generating addressing code for scalar variables and quantify the improvements due to optimizations such as offset assignment, modify register optimization and address register assignment. We propose an offset assignment heuristic that uses k address registers, an optimal dynamic programming algorithm for modify register optimization, and an optimal formulation and a heuristic algorithm for the address register assignment problem.

Cost minimization and workload balancing in printed circuit board assembly

Abstract: We consider an operation assignment problem arising from a Printed Circuit (PC) board assembly process. The research was inspired by applications at Hewlett-Packard Company where hundreds of types of PC boards require the insertion of thousands of types of components. The components can be inserted manually or by automated insertion machines. The machines can only hold a limited number of different component types. We investigate how to assign the boards and components to the machines and manual process so as to minimize cost while at the same time balancing machine workloads. We first present a Binary Integer Program (BIP) formulation of the problem. We then develop optimality results that allow us to reduce significantly the size of the BIP. Using the improved BIP formulation, and upper bounds generated using a Cost Minimizing Workload Balancing (CMWB) heuristic that we develop, we show how branch-and-bound can be used to find optimal solutions to small and medium-sized problems in reasonable time. We a...

Multiple Hotlink Assignment

Abstract: The input for the hotlink assignment problem consists of a node weighted directed acyclic graph with a designated root node r. The goal is to minimize the weighted shortest path length rooted at r by adding a restricted number of outgoing arcs (hotlinks) to each node. The (h, k)-hotlink assignment problem is defined on k-regular complete trees, and at most h hotlinks can be assigned to each node. We contribute algorithms for the (1, k), (2, k), and (k-1, k) hotlink assignment problem.

Robust pole assignment in descriptor linear systems via state feedback

Abstract: The problem of eigenvalue assignment with minimum sensitivity in multivariable descriptor linear systems via state feedback is considered. Based on the perturbation theory of generalized eigenvalues of matrix pairs, the sensitivity measures of the closed-loop finite eigenvalues are established in terms of the closed-loop normalized right and left eigenvectors. By combining these measures with a recently proposed general parametric eigenstructure assignment result for descriptor linear systems via state feedback, the robust pole assignment problem is converted into an independent minimization problem. The optimality of the obtained solution to the robust pole assignment problem is totally dependent on the solution to the independent minimization problem. The closed-loop eigenvalues are also taken as a part of the design parameters and are optimized, together with the other degrees of freedom, within certain desired regions on the complex plane. The approach takes numerical stability into consideration and also gives good robustness for the closed-loop regularity.