Network Flows: Theory, Algorithms, and ApplicationsA comprehensive introduction to network flows that brings together the classic and the contemporary aspects of the field, and provides an integrative view of theory, algorithms, and applications.
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Page 308
... reduced cost " for arc ( i , j ) in the sense that it measures the cost of this arc relative to the shortest path distances d ( i ) and d ( j ) . Notice that with respect to the optimal distances , every arc in the network has a ...
... reduced cost " for arc ( i , j ) in the sense that it measures the cost of this arc relative to the shortest path distances d ( i ) and d ( j ) . Notice that with respect to the optimal distances , every arc in the network has a ...
Page 309
... costs . We can now provide an alternative form of the negative cycle optimality conditions , stated in terms of the reduced costs of the arcs . Theorem 9.3 ( Reduced Cost Optimality Conditions ) . A feasible solution x * is an optimal ...
... costs . We can now provide an alternative form of the negative cycle optimality conditions , stated in terms of the reduced costs of the arcs . Theorem 9.3 ( Reduced Cost Optimality Conditions ) . A feasible solution x * is an optimal ...
Page 339
... Cost Sensitivity Analysis Finally , we discuss changes in arc costs , which we assume are integral . We discuss the ... reduced cost of arc ( p , q ) by 1 unit as well . If c < 0 before the change , then after the change , the modified ...
... Cost Sensitivity Analysis Finally , we discuss changes in arc costs , which we assume are integral . We discuss the ... reduced cost of arc ( p , q ) by 1 unit as well . If c < 0 before the change , then after the change , the modified ...
Contents
INTRODUCTION | 1 |
PATHS TREES AND CYCLES | 23 |
ALGORITHM DESIGN AND ANALYSIS | 53 |
Copyright | |
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adjacency list Application arc costs arc flows arc lengths augmenting path augmenting path algorithm bipartite capacity scaling algorithm commodity constraints cost flow problem define denote Dijkstra's algorithm directed path discussion distance label example Exercise feasible flow feasible solution Fibonacci heap flow algorithms formulation implementation integer iteration label-correcting algorithm Lagrangian multiplier Lagrangian relaxation Lemma linear programming lower bound matching matrix maximum flow problem minimum cost flow minimum spanning tree multicommodity flow problem N₁ negative cycle network contains network flow problem network G network simplex algorithm node potentials nonnegative NP-complete O(nm objective function value operation optimal solution optimality conditions path from node polynomial preflow-push algorithm reduced cost residual network s-t cut satisfies scaling phase Section shortest path distances shortest path problem Show shown in Figure simplex method source node Suppose Theorem undirected units of flow upper bound variables zero