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77
Geometric Shortest Paths and Network Optimization
 Handbook of Computational Geometry
, 1998
"... Introduction A natural and wellstudied problem in algorithmic graph theory and network optimization is that of computing a "shortest path" between two nodes, s and t, in a graph whose edges have "weights" associated with them, and we consider the "length" of a path to ..."
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Cited by 196 (15 self)
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Introduction A natural and wellstudied problem in algorithmic graph theory and network optimization is that of computing a "shortest path" between two nodes, s and t, in a graph whose edges have "weights" associated with them, and we consider the "length" of a path to be the sum of the weights of the edges that comprise it. Efficient algorithms are well known for this problem, as briefly summarized below. The shortest path problem takes on a new dimension when considered in a geometric domain. In contrast to graphs, where the encoding of edges is explicit, a geometric instance of a shortest path problem is usually specified by giving geometric objects that implicitly encode the graph and its edge weights. Our goal in devising efficient geometric algorithms is generally to avoid explicit construction of the entire underlying graph, since the full induced graph may be very large (even exponential in the input size, or infinite). Computing an optimal
MultiConstrained Optimal Path Selection
, 2001
"... Providing qualityofservice (QoS) guarantees in packet networks gives rise to several challenging issues. One of them is how to determine a feasible path that satisfies a set of constraints while maintaining high utilization of network resources. The latter objective implies the need to impose an a ..."
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Cited by 76 (1 self)
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Providing qualityofservice (QoS) guarantees in packet networks gives rise to several challenging issues. One of them is how to determine a feasible path that satisfies a set of constraints while maintaining high utilization of network resources. The latter objective implies the need to impose an additional optimality requirement on the feasibility problem. This can be done through a primary cost function (e.g., administrative weight, hopcount) according to which the selected feasible path is optimal. In general, multiconstrained path selection, with or without optimization, is an NPcomplete problem that cannot be exactly solved in polynomial time. Heuristics and approximation algorithms with polynomialand pseudopolynomialtime complexities are often used to deal with this problem. However, existing solutions suffer either from excessive computational complexities that cannot be used for online network operation or from low performance. Moreover, they only deal with special cases of the problem (e.g., two constraints without optimization, one constraint with optimization, etc.). For the feasibility problem under multiple constraints, some researchers have recently proposed a nonlinear cost function whose minimization provides a continuous spectrum of solutions ranging from a generalized linear approximation (GLA) to an asymptotically exact solution. In this paper, we propose an efficient heuristic algorithm for the most general form of the problem. We first formalize the theoretical properties of the above nonlinear cost function. We then introduce our heuristic algorithm (H MCOP), which attempts to minimize both the nonlinear cost function (for the feasibility part) and the primary cost function (for the optimality part). We prove that H MCOP guarantees at least t...
Shortest Path Algorithms in Transportation Models: Classical and Innovative Aspects
, 1998
"... Shortest Path Problems are among the most studied network flow optimization problems, with interesting applications in various fields. One such field is transportation, where shortest path problems of different kinds need to be solved. Due to the nature of the application, transportation scientists ..."
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Cited by 67 (3 self)
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Shortest Path Problems are among the most studied network flow optimization problems, with interesting applications in various fields. One such field is transportation, where shortest path problems of different kinds need to be solved. Due to the nature of the application, transportation scientists need very flexible and efficient shortest path procedures, both from the running time point of view, and also for the memory requirements. Since no "best" algorithm currently exists for every kind of transportation problem, research in this field has recently moved to the design and implementation of "ad hoc" shortest path procedures, which are able to capture the peculiarities of the problems under consideration. The aim of this work is to present in a unifying framework both the main algorithmic approaches that have been proposed in the past years for solving the shortest path problems arising most frequently in the transportation field, and also some important implementation techniques ...
An orthogonal genetic algorithm for multimedia multicast routing
 IEEE Trans. Evolutionary Computation
"... Abstract—Many multimedia communication applications require a source to send multimedia information to multiple destinations through a communication network. To support these applications, it is necessary to determine a multicast tree of minimal cost to connect the source node to the destination n ..."
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Cited by 36 (6 self)
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Abstract—Many multimedia communication applications require a source to send multimedia information to multiple destinations through a communication network. To support these applications, it is necessary to determine a multicast tree of minimal cost to connect the source node to the destination nodes subject to delay constraints on multimedia communication. This problem is known as multimedia multicast routing and has been proved to be NPcomplete. This paper proposes an orthogonal genetic algorithm for multimedia multicast routing. Its salient feature is to incorporate an experimental design method called orthogonal design into the crossover operation. As a result, it can search the solution space in a statistically sound manner and it is well suited for parallel implementation and execution. We execute the orthogonal genetic algorithm to solve two sets of benchmark test problems. The results indicate that for practical problem sizes, the orthogonal genetic algorithm can find nearoptimal solutions within moderate numbers of generations. Index Terms—Genetic algorithm, multimedia multicast routing, NPcomplete. I.
The Quickest Multicommodity Flow Problem
 INTEGER PROGRAMMING AND COMBINATORIAL OPTIMIZATION
, 2002
"... Traditionally, ows over time are solved in timeexpanded networks which contain one copy of the original network for each discrete time step. While this method makes available the whole algorithmic toolbox developed for static ows, its main and often fatal drawback is the enormous size of the tim ..."
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Cited by 35 (11 self)
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Traditionally, ows over time are solved in timeexpanded networks which contain one copy of the original network for each discrete time step. While this method makes available the whole algorithmic toolbox developed for static ows, its main and often fatal drawback is the enormous size of the timeexpanded network. In particular, this approach usually does not lead to efficient algorithms with running time polynomial in the input size since the size of the timeexpanded network is only pseudopolynomial. We present two
An Efficient Algorithm for Finding a Path Subject to Two Additive Constraints
 Computer Communications Journal
, 2000
"... One of the key issues in providing endtoend qualityofservice (QoS) guarantees in packet networks is how to determine a feasible route that satisfies a set of constraints. In general, finding a path subject to multiple additive constraints (e.g., delay, delayjitter) is an NPcomplete problem ..."
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Cited by 35 (5 self)
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One of the key issues in providing endtoend qualityofservice (QoS) guarantees in packet networks is how to determine a feasible route that satisfies a set of constraints. In general, finding a path subject to multiple additive constraints (e.g., delay, delayjitter) is an NPcomplete problem that cannot be exactly solved in polynomial time. Accordingly, several heuristics and approximation algorithms have been proposed for this problem. Many of these algorithms suffer from either excessive computational cost or low performance. In this paper, we provide an efficient approximation algorithm for finding a path subject to two additive constraints. The worstcase computational complexity of this algorithm is within a logarithmic number of calls to Dijkstra's shortest path algorithm. Its average complexity is even much lower than that, as demonstrated by simulation experiments. The performance of the proposed algorithm is justified via theoretical bounds that are provided for ...
Resource Constrained Shortest Paths
 PROC. 8TH EUROPEAN SYMPOSIUM ON ALGORITHMS (ESA), SPRINGER LNCS
, 2000
"... The resource constrained shortest path problem (CSP) asks for the computation of a least cost path obeying a set of resource constraints. The problem is NPcomplete. We give theoretical and experimental results for CSP. In the theoretical part we present the hull approach, a combinatorial algorit ..."
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Cited by 33 (1 self)
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The resource constrained shortest path problem (CSP) asks for the computation of a least cost path obeying a set of resource constraints. The problem is NPcomplete. We give theoretical and experimental results for CSP. In the theoretical part we present the hull approach, a combinatorial algorithm for solving a linear programming relaxation and prove that it runs in polynomial time in the case of one resource. In the experimental part we compare the hull approach to previous methods for solving the LP relaxation and give an exact algorithm based on the hull approach. We also compare our exact algorithm to previous exact algorithms and approximation algorithms for the problem.
An approximate algorithm for combinatorial optimization problems with two parameters
 Australasian Journal of Combinatorics
, 1996
"... with two parameters ..."
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Quickest Flows Over Time
 SIAM Journal on Computing
, 2003
"... Flows over time (also called dynamic flows) generalize standard network flows by introducing an element of time. They naturally model problems where travel and transmission are not instantaneous. Traditionally, flows over time are solved in timeexpanded networks that contain one copy of the orig ..."
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Cited by 27 (9 self)
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Flows over time (also called dynamic flows) generalize standard network flows by introducing an element of time. They naturally model problems where travel and transmission are not instantaneous. Traditionally, flows over time are solved in timeexpanded networks that contain one copy of the original network for each discrete time step. While this method makes available the whole algorithmic toolbox developed for static flows, its main and often fatal drawback is the enormous size of the timeexpanded network. We present several approaches for coping with this difficulty.
Lagrangian Relaxation and Enumeration for Solving Constrained
 ShortestPath Problems, Operations Research Department, Naval Postgraduate School
, 2003
"... Recently published research indicates that a vertexlabeling algorithm based on dynamicprogramming concepts is the most efficient procedure available for solving constrained shortestpath problems (CSPPs), i.e., shortestpath problems with one or more side constraints on the total “weight ” of the ..."
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Cited by 23 (6 self)
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Recently published research indicates that a vertexlabeling algorithm based on dynamicprogramming concepts is the most efficient procedure available for solving constrained shortestpath problems (CSPPs), i.e., shortestpath problems with one or more side constraints on the total “weight ” of the optimal path. However, we investigate an alternative procedure that Lagrangianises the side constraints, optimises the resulting Lagrangian function and then closes any duality gap through enumeration of nearshortest paths. These paths are measured with respect to Lagrangianmodified edge lengths, and “nearshortest ” implies ɛoptimal, with ɛ equal to the duality gap. Our recently developed procedure for enumerating nearshortestpaths leads to an algorithm for CSPP that, empirically, proves to be an order of magnitude faster than the most recent vertexlabeling algorithm. 1