Ravindra K. Ahuja

Faster algorithms for the shortest path problem

Efficient implementations of Dijkstras shortest path algorithm are investigated. A new data structure, called the <italic>radix heap</italic>, is proposed for use in this algorithm. On a network with <italic>n</italic> vertices, <italic>m</italic> edges, and nonnegative integer arc costs bounded by <italic>C</italic>, a one-level form of radix heap gives a time bound for Dijkstras algorithm of <italic>O</italic>(<italic>m</italic> + <italic>n</italic> log <italic>C</italic>). A two-level form of radix heap gives a bound of <italic>O</italic>(<italic>m</italic> + <italic>n</italic> log <italic>C</italic>/log log <italic>C</italic>). A combination of a radix heap and a previously known data structure called a <italic>Fibonacci heap</italic> gives a bound of <italic>O</italic>(<italic>m</italic> + <italic>n</italic>a @@@@log <italic>C</italic>). The best previously known bounds are <italic>O</italic>(<italic>m</italic> + <italic>n</italic> log <italic>n</italic>) using Fibonacci heaps alone and <italic>O</italic>(<italic>m</italic> log log <italic>C</italic>) using the priority queue structure of Van Emde Boas et al. [ 17].

A greedy genetic algorithm for the quadratic assignment problem

Abstract The Quadratic Assignment Problem (QAP) is one of the classical combinatorial optimization problems and is known for its diverse applications. In this paper, we suggest a genetic algorithm for the QAP and report its computational behavior. The genetic algorithm incorporates many greedy principles in its design and, hence, we refer to it as a greedy genetic algorithm. The ideas we incorporate in the greedy genetic algorithm include (i) generating the initial population using a randomized construction heuristic; (ii) new crossover schemes; (iii) a special purpose immigration scheme that promotes diversity; (iv) periodic local optimization of a subset of the population; (v) tournamenting among different populations; and (vi) an overall design that attempts to strike a balance between diversity and a bias towards fitter individuals. We test our algorithm on all the benchmark instances of QAPLIB, a well-known library of QAP instances. Out of the 132 total instances in QAPLIB of varied sizes, the greedy genetic algorithm obtained the best known solution for 103 instances, and for the remaining instances (except one) found solutions within 1% of the best known solutions. Scope and purpose Genetic Algorithms (GAs) are one of the most popular heuristic algorithms to solve optimization problems and is an extremely useful tool in an OR toolkit. For solving combinatorial optimization problems, GAs in their elementary forms are not competitive with other heuristic algorithms such as simulated annealing and tabu serach. In this paper, we have investigated the use of several possible enhancements to GAs and illustrated them using the Quadratic Assignment Problem (QAP), one of the hardest nut in the field of combinatorial optimization. Most of our enhancements use some greedy criteria to improve the quality of individuals in the population. We found that the overall performance of the GA for the QAP improves by using greedy methods but not their overuse. Overuse of greedy methods adversely affects the diversity in the population. By striking a right balance between the greedy methods that improve the quality of the solution and the methods that promote diversity, we can obtain fairly effective heuristic algorithms for solving combinatorial optimization problems.

Inverse Optimization

In this paper, we study inverse optimization problems defined as follows. LetS denote the set of feasible solutions of an optimization problemP, letc be a specified cost vector, andx0 be a given feasible solution. The solutionx0 may or may not be an optimal solution ofP with respect to the cost vectorc. The inverse optimization problem is to perturb the cost vectorc tod so thatx0 is an optimal solution ofP with respect tod and ||d- c || p is minimum, where ||d- c || p is some selectedLp norm. In this paper, we consider the inverse linear programming problem underL1 norm (where ||d- c || p= ? j?Jw j|d j-c j|, withJ denoting the index set of variablesx jandw jdenoting the weight of the variablej) and underL8 norm (where||d- c || p= max j?J{w j|d j-c j|} ). We prove the following results: (i) If the problemP is a linear programming problem, then its inverse problem under theL1 as well asL8 norm is also a linear programming problem. (ii) If the problemP is a shortest path, assignment or minimum cut problem, then its inverse problem under theL1 norm and unit weights can be solved by solving a problem of the same kind. For the nonunit weight case, the inverse problem reduces to solving a minimum cost flow problem. (iii) If the problemP is a minimum cost flow problem, then its inverse problem under theL1 norm and unit weights reduces to solving a unit-capacity minimum cost flow problem. For the nonunit weight case, the inverse problem reduces to solving a minimum cost flow problem. (iv) If the problemP is a minimum cost flow problem, then its inverse problem under theL8 norm and unit weights reduces to solving a minimum mean cycle problem. For the nonunit weight case, the inverse problem reduces to solving a minimum cost-to-time ratio cycle problem. (v) If the problemP is polynomially solvable for linear cost functions, then inverse versions ofP under theL1 andL8 norms are also polynomially solvable.

A novel linear programming approach to fluence map optimization for intensity modulated radiation therapy treatment planning

We present a novel linear programming (LP) based approach for efficiently solving the intensity modulated radiation therapy (IMRT) fluence-map optimization (FMO) problem to global optimality. Our model overcomes the apparent limitations of a linear-programming approach by approximating any convex objective function by a piecewise linear convex function. This approach allows us to retain the flexibility offered by general convex objective functions, while allowing us to formulate the FMO problem as a LP problem. In addition, a novel type of partial-volume constraint that bounds the tail averages of the differential dose-volume histograms of structures is imposed while retaining linearity as an alternative approach to improve dose homogeneity in the target volumes, and to attempt to spare as many critical structures as possible. The goal of this work is to develop a very rapid global optimization approach that finds high quality dose distributions. Implementation of this model has demonstrated excellent results. We found globally optimal solutions for eight 7-beam head-and-neck cases in less than 3 min of computational time on a single processor personal computer without the use of partial-volume constraints. Adding such constraints increased the running times by a factor of 2-3, but improved the sparing of critical structures. All cases demonstrated excellent target coverage (> 95%), target homogeneity (< 10% overdosing and < 7% underdosing) and organ sparing using at least one of the two models.

A New Linear Programming Approach to Radiation Therapy Treatment Planning Problems

We consider the problem of radiation therapy treatment planning for cancer patients. During radiation therapy, beams of radiation pass through a patient, killing both cancerous and normal cells. Thus, the radiation therapy must be carefully planned so that a clinically prescribed dose is delivered to targets containing cancerous cells, while nearby organs and tissues are spared. Currently, a technique called intensity-modulated radiation therapy (IMRT) is considered to be the most effective radiation therapy for many forms of cancer. In IMRT, the patient is irradiated from several beams, each of which is decomposed into hundreds of small beamlets, the intensities of which can be controlled individually. In this paper, we consider the problem of designing a treatment plan for IMRT when the orientations of the beams are given. We propose a new model that has the potential to achieve most of the goals with respect to the quality of a treatment plan that have been considered to date. However, in contrast with established mixed-integer and nonlinear programming formulations, we do so while retaining linearity of the optimization problem, which substantially improves the tractability of the optimization problem. Furthermore, we discuss how several additional quality and practical aspects of the problem that have been ignored to date can be incorporated into our linear model. We demonstrate the effectiveness of our approach on clinical data.

A Column Generation Approach to Radiation Therapy Treatment Planning Using Aperture Modulation

This paper considers the problem of radiation therapy treatment planning for cancer patients. During radiation therapy, beams of radiation pass through a patient. This radiation kills both cancerous and normal cells, so the radiation therapy must be carefully planned to deliver a clinically prescribed dose to certain targets while sparing nearby organs and tissues. Currently, a technique called intensity modulated radiation therapy (IMRT) is considered to be the most effective radiation therapy for many forms of cancer. In IMRT, the patient is irradiated from several different directions. From each direction, one or more irregularly shaped radiation beams of uniform intensity are used to deliver the treatment. This paper deals with the problem of designing a treatment plan for IMRT that determines an optimal set of such shapes (called apertures) and their corresponding intensities. This is in contrast with established two-stage approaches where, in the first phase, each radiation beam is viewed as consisting of a set of individual beamlets, each with its own intensity. A second phase is then needed to approximate and decompose the optimal intensity profile into a set of apertures with corresponding intensities. The problem is formulated as a large-scale convex programming problem, and a column generation approach to deal with its dimensionality is developed. The associated pricing problem determines, in each iteration, one or more apertures to be added to our problem. Several variants of this pricing problem are discussed, each corresponding to a particular set of constraints that the apertures must satisfy in one or more of the currently available types of commercial IMRT equipment. Polynomial-time algorithms are presented for solving each of these variants of the pricing problem to optimality. Finally, the effectiveness of our approach is demonstrated on clinical data.

Multi-exchange neighborhood structures for the capacitated minimum spanning tree problem

Abstract.The capacitated minimum spanning tree (CMST) problem is to find a minimum cost spanning tree with an additional cardinality constraint on the sizes of the subtrees incident to a given root node. The CMST problem is an NP-complete problem, and existing exact algorithms can solve only small size problems. Currently, the best available heuristic procedures for the CMST problem are tabu search algorithms due to Amberg et al. and Sharaiha et al. These algorithms use two-exchange neighborhood structures that are based on exchanging a single node or a set of nodes between two subtrees. In this paper, we generalize their neighborhood structures to allow exchanges of nodes among multiple subtrees simultaneously; we refer to such neighborhood structures as multi-exchange neighborhood structures. Our first multi-exchange neighborhood structure allows exchanges of single nodes among several subtrees. Our second multi-exchange neighborhood structure allows exchanges that involve multiple subtrees. The size of each of these neighborhood structures grows exponentially with the problem size without any substantial increase in the computational times needed to find improved neighbors. Our approach, which is based on the cyclic transfer neighborhood structure due to Thompson and Psaraftis and Thompson and Orlin transforms a profitable exchange into a negative cost subset-disjoint cycle in a graph, called an improvement graph, and identifies these cycles using variants of shortest path label-correcting algorithms. Our computational results with GRASP and tabu search algorithms based on these neighborhood structures reveal that (i) for the unit demand case our algorithms obtained the best available solutions for all benchmark instances and improved some; and (ii) for the heterogeneous demand case our algorithms improved the best available solutions for most of the benchmark instances with improvements by as much as 18%. The running times our multi-exchange neighborhood search algorithms are comparable to those taken by two-exchange neighborhood search algorithms.

Finding Minimum-Cost Flows by Double-Scaling

Several researchers have recently developed new techniques that give fast algorithms for the minimum-cost flow problem. In this paper we combine several of these techniques to yield an algorithm running in O(nm(log logU) log(nC)) time on networks withn vertices,m edges, maximum arc capacityU, and maximum arc cost magnitudeC. The major techniques used are the capacity-scaling approach of Edmonds and Karp, the excess-scaling approach of Ahuja and Orlin, the cost-scaling approach of Goldberg and Tarjan, and the dynamic tree data structure of Sleator and Tarjan. For nonsparse graphs with large maximum arc capacity, we obtain a similar but slightly better bound. We also obtain a slightly better bound for the (noncapacitated) transportation problem. In addition, we discuss a capacity-bounding approach to the minimum-cost flow problem.

Improved time bounds for the maximum flow problem

Recently, Goldberg proposed a new approach to the maximum network flow problem. The approach yields a very simple algorithm running in

Solving Real-Life Railroad Blocking Problems

O(n^3 )