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A Faster Algorithm for the Inverse Spanning Tree Problem
 JOURNAL OF ALGORITHMS
, 2000
"... In this paper, we consider the inverse spanning tree problem. Given an undirected graph G 0 = (N 0 , A 0 ) with n nodes, m arcs, an arc cost vector c, and a spanning tree T 0 , the inverse spanning tree problem is to perturb the arc cost vector c to a vector d so that T 0 is a minimum span ..."
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Cited by 6 (2 self)
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In this paper, we consider the inverse spanning tree problem. Given an undirected graph G 0 = (N 0 , A 0 ) with n nodes, m arcs, an arc cost vector c, and a spanning tree T 0 , the inverse spanning tree problem is to perturb the arc cost vector c to a vector d so that T 0 is a minimum spanning tree with respect to the cost vector d and the cost of perturbation given by d  c =  d c  ij ij (i, j) A  is minimum. We show that the dual of the inverse spanning tree problem is a bipartite node weighted matching problem on a specially structured graph (which we call the path graph) that contains m nodes and as many as (mn+1)(n1) = O(nm) arcs. We first transform the bipartite node weighted matching problem into a specially structured minimum cost flow problem and use its special structure to develop an O(n³) algorithm. We next use its special structure more effectively and develop an O(n² log n) time algorithm. This improves the previous O(n³) time algorithm due to Sokkalingam, Ahuja and Orlin [1999].
Inapproximability results for the inverse shortest paths problem with integer lengths and unique shortest paths
 in Proceedings of the Second International Network Optimization Conference(INOC 2005
, 2005
"... We study the complexity of two Inverse Shortest Paths (ISP) problems with integer arc lengths and the requirement for uniquely determined shortest paths. Given a collection of paths in a directed graph, the task is to find positive integer arc lengths such that the given paths are uniquely determine ..."
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Cited by 3 (2 self)
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We study the complexity of two Inverse Shortest Paths (ISP) problems with integer arc lengths and the requirement for uniquely determined shortest paths. Given a collection of paths in a directed graph, the task is to find positive integer arc lengths such that the given paths are uniquely determined shortest paths between their respective terminals. The first problem seeks for arc lengths that minimize the length of the longest of the prescribed paths. In the second problem, the length of the longest arc is to be minimized. We show that it is NPhard to approximate the minimal longest path length within a factor less than 8/7 or the minimal longest arc length within a factor less than 9/8. This answers the (previously) open question whether these problems are NPhard or not. We also present a simple algorithm that achieves an O(V )approximation guarantee for both variants. Both ISP problems arise in the planning of telecommunication networks with shortest path routing protocols. Our results imply that it is NPhard to decide whether a given path set can be realized with a real shortest path routing protocol such as OSPF, ISIS, or RIP.
Finding Small Administrative Lengths for Shortest Path Routing
 INOC 2005, Book
, 2005
"... We study the problem of finding small integer arc lengths in a digraph such that some given paths are uniquely determined shortest paths. This Inverse Shortest Paths (ISP) problem arises in the planning of shortest path networks. We present an O(V )approximation algorithm for ISP and show that mi ..."
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Cited by 1 (0 self)
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We study the problem of finding small integer arc lengths in a digraph such that some given paths are uniquely determined shortest paths. This Inverse Shortest Paths (ISP) problem arises in the planning of shortest path networks. We present an O(V )approximation algorithm for ISP and show that minimizing the longest arc or the longest path length is APXhard. Thus, deciding if a given path set can be realized with a real shortest path routing protocol such as OSPF, ISIS, or RIP is N Phard.
Imputing a Convex Objective Function
"... Abstract — We consider an optimizing process (or parametric optimization problem), i.e., an optimization problem that depends on some parameters. We present a method for imputing or estimating the objective function, based on observations of optimal or nearly optimal choices of the variable for seve ..."
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Cited by 1 (1 self)
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Abstract — We consider an optimizing process (or parametric optimization problem), i.e., an optimization problem that depends on some parameters. We present a method for imputing or estimating the objective function, based on observations of optimal or nearly optimal choices of the variable for several values of the parameter, and prior knowledge (or assumptions) about the objective. Applications include estimation of consumer utility functions from purchasing choices, estimation of value functions in control problems, given observations of an optimal (or just good) controller, and estimation of cost functions in a flow network. I.