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A Linear Algorithm for Analysis of Minimum Spanning and Shortest Path Trees of Planar Graphs
 Algorithmica
, 1992
"... We give a linear time and space algorithm for analyzing trees in planar graphs. The algorithm can be used to analyze the sensitivity of a minimum spanning tree to changes in edge costs, to find its replacement edges, and to verify its minimality. It can also be used to analyze the sensitivity of a s ..."
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We give a linear time and space algorithm for analyzing trees in planar graphs. The algorithm can be used to analyze the sensitivity of a minimum spanning tree to changes in edge costs, to find its replacement edges, and to verify its minimality. It can also be used to analyze the sensitivity of a singlesource shortest path tree to changes in edge costs, and to analyze the sensitivity of a minimum cost network flow. The algorithm is simple and practical. It uses the properties of a planar embedding, combined with a heapordered queue data structure. Let G = (V; E) be a planar graph, either directed or undirected, with n vertices and m = O(n) edges. Each edge e 2 E has a realvalued cost cost(e). A minimum spanning tree of a connected, undirected planar graph G is a spanning tree of minimum total edge cost. If G is directed and r is a vertex from which all other vertices are reachable, then a shortest path tree from r is a spanning tree that contains a minimumcost path from r to every...
Complexity of determining exact tolerances for minmax combinatorial optimization problems
 University of Groningen
, 2000
"... SOMtheme A Primary Processes within Firms Suppose that we are given an instance of a combinatorial optimization problem with minmax objective along with an optimal solution for it. Let the cost of a single element be varied. We refer to the range of values of the element’s cost for which the given ..."
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SOMtheme A Primary Processes within Firms Suppose that we are given an instance of a combinatorial optimization problem with minmax objective along with an optimal solution for it. Let the cost of a single element be varied. We refer to the range of values of the element’s cost for which the given optimal solution remains optimal as its exact tolerance. In this paper we examine the problem of determining the exact tolerance of each element in combinatorial optimization problems with minmax objectives. We show that under very weak assumptions, the exact tolerance of each element can be determined in polynomial time if and only if the original optimization problem can be solved in polynomial time.
3 Concepts of Stability in Discrete Optimization Involving Generalized Addition Operations
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optimization with an additive objective function
"... Extremal values of global tolerances in combinatorial ..."
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