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Constrained Shortest LinkDisjoint Paths Selection: A Network Programming Based Approach
"... Given a graph G with each link in the graph associated with two positive weights, cost and delay, we consider the problem of selecting a set of k linkdisjoint paths from a node s to another node t such that the total cost of these paths is minimum and that the total delay of these paths is not grea ..."
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Cited by 5 (1 self)
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Given a graph G with each link in the graph associated with two positive weights, cost and delay, we consider the problem of selecting a set of k linkdisjoint paths from a node s to another node t such that the total cost of these paths is minimum and that the total delay of these paths is not greater than a specified bound. This problem, to be called the CSDP(k) problem, can be formulated as an integer linear programming (ILP) problem. Relaxing the integrality constraints results in an upper bounded linear programming problem. We first show that the integer relaxations of the CSDP(k) problem and a generalized version of this problem to be called the GCSDP(k) problem (in which each path is required to satisfy a specified bound on its delay) both have the same optimal objective value. In view of this we focus our work on the relaxed form of the CSDP(k) problem called RELAXCSDP(k). We study RELAXCSDP(k) from the primal perspective using the revised simplex method of linear programming. We discuss different issues such as formulas to identify entering and leaving variables, anticycling strategy, computational time complexity etc., related to an efficient implementation of our approach. We show how to extract from an optimal solution to RELAXCSDP(k) a set of k linkdisjoint st paths which is an approximate solution to the original CSDP(k) problem. We also derive bounds on the quality of this solution with respect to the optimum. We present simulation results that demonstrate that our algorithm is faster than currently available approaches. Our simulation results also indicate that in most cases the individual delays of the paths produced starting from RELAXCSDP(k) do not deviate in a significant way from the individual path delay requirements of the GCSDP(k) problem.
QoS routing in communication networks: Approximation algorithms based on the primal simplex method of linear programming
 IEEE TRANSACTIONS ON COMPUTERS
, 2006
"... Given a directed network with two integer weights, cost and delay, associated with each link, QualityofService (QoS) routing requires the determination of a minimum cost path from one node to another node such that the delay of the path is bounded by a specified integer value. This problem, also k ..."
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Cited by 3 (2 self)
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Given a directed network with two integer weights, cost and delay, associated with each link, QualityofService (QoS) routing requires the determination of a minimum cost path from one node to another node such that the delay of the path is bounded by a specified integer value. This problem, also known as the Constrained Shortest Path problem (CSP), admits an Integer Linear Programming (ILP) formulation. Due to the integrality constraints, the problem is NPhard. So, approximation algorithms have been presented in the literature. Among these, the LARAC algorithm, based on the dual of the LP relaxation of the CSP problem, is very efficient. In contrast to most of the currently available approaches, we study this problem from a primal perspective. Several issues relating to efficient implementations of our approach are discussed. We present two algorithms of pseudopolynomial time complexity. One of these allows degenerate pivots and uses an anticycling strategy and the other, called the NBS algorithm, is based on a novel strategy which avoids degenerate pivots. Experimental results comparing the NBS algorithm, the LARAC algorithm, and general purpose LP solvers are presented. In all cases, the NBS algorithm compares favorably with others and beats them on dense networks.
A Lagrangian Decomposition/Evolutionary Algorithm Hybrid for the Knapsack Constrained Maximum Spanning Tree Problem
, 2007
"... We present a Lagrangian decomposition approach for the Knapsack Constrained Maximum Spanning Tree problem yielding upper bounds as well as heuristic solutions. This method is further combined with an evolutionary algorithm to a sequential hybrid approach. Thorough experimental investigations, incl ..."
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We present a Lagrangian decomposition approach for the Knapsack Constrained Maximum Spanning Tree problem yielding upper bounds as well as heuristic solutions. This method is further combined with an evolutionary algorithm to a sequential hybrid approach. Thorough experimental investigations, including a comparison to a previously suggested simpler Lagrangian relaxation based method, document the advantages of our approach. Most of the upper bounds derived by Lagrangian decomposition are optimal, and when additionally applying local search (LS) and combining it with the evolutionary algorithm, large and supposedly hard instances can be either solved to provable optimality or with a very small remaining gap in reasonable time.
Simultaneous Solution of Lagrangean . . Preprocessing for the Weight Constrained Shortest Path Problem
, 2007
"... Conventional Lagrangean preprocessing for the network Weight Constrained Shortest Path Christofides [3], calculates lower bounds on the cost of using each node and edge in a feasible path using a single optimal Lagrange multiplier for the relaxation of the WCSPP. These lower bounds are used in conj ..."
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Conventional Lagrangean preprocessing for the network Weight Constrained Shortest Path Christofides [3], calculates lower bounds on the cost of using each node and edge in a feasible path using a single optimal Lagrange multiplier for the relaxation of the WCSPP. These lower bounds are used in conjunction with an upper bound to eliminate nodes and edges. However, for each node and edge, a Lagrangean dual problem exists whose solution may differ from the relaxation of the full problem. Thus, using one Lagrange multiplier does not offer the best possible network reduction. Furthermore, eliminating nodes and edges from the network may change the Lagrangean dual solutions in the remaining reduced network, warranting an iterative solution and reduction procedure. We develop a method for solving the related Lagrangean dual problems for each edge simultaneously which is iterated with eliminating nodes and edges. We demonstrate the effectiveness of our method computationally: we test it against several others and show that it both reduces solve time and the number of intractable problems encountered. We use a modified version of Carlyle and Wood’s [6] enumeration algorithm in the gap closing stage. We also make improvements to this algorithm and test them computationally.