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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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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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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.
Complexity of some inverse shortest path lengths problems,” Networks
"... The input to an inverse shortest path lengths problem (ISPL) consists of a graph G with arc weights, and a collection of sourcesink pairs with prescribed distances that do not necessarily conform to the shortest path lengths in G. The goal is to modify the arc weights, subject to a penalty on the d ..."
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The input to an inverse shortest path lengths problem (ISPL) consists of a graph G with arc weights, and a collection of sourcesink pairs with prescribed distances that do not necessarily conform to the shortest path lengths in G. The goal is to modify the arc weights, subject to a penalty on the deviation from the given weights, so that the shortest path lengths are equal to the prescribed values. We show that although ISPL is an NPhard problem, several ISPL classes are polynomially solvable. These cases include ISPL where the collection of the pairs share a single source and all other nodes as destinations (the singlesource allsink problem SAISPL). For the case where the collection contains a single node pair (the singlesource singlesink problem SSISPL), we identify conditions on the uniformity of the penalty functions and on the original arc weights, which make SSISPL polynomially solvable. These results cannot be strengthened significantly as the general singlesource ISPL is NPhard and the allsink case, with more than one source, is also NPhard. We further provide a convex programming formulation for a relaxation of ISPL in which the shortest path lengths are only required to be no less than the given values (LBISPL). It is demonstrated how this compact formulation leads to efficient algorithms for ISPL.
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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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.
On the Inverse Shortest Paths Problem
"... In this paper we survey a variant of the inverse shortest paths problem. Given is a network in which certain nodepairs require connections over specific predefined single paths. The aim is to find link weights such that the desired paths are uniquely induced by a shortestpath algorithm. Further we ..."
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In this paper we survey a variant of the inverse shortest paths problem. Given is a network in which certain nodepairs require connections over specific predefined single paths. The aim is to find link weights such that the desired paths are uniquely induced by a shortestpath algorithm. Further we require that the link weights fit existing shortestpath based routing protocols. We give a motivation for this problem, discuss a number of its solvability aspects, and show how a feasible solution can be obtained from its linear relaxation. A network operator who wants to obtain a specific routing pattern by adjusting link weights, will find practical use of the solution to the considered problem. 1
THE INVERSE MAXIMUM FLOW PROBLEM WITH LOWER AND UPPER BOUNDS FOR THE FLOW
, 2006
"... Abstract: The general inverse maximum flow problem (denoted GIMF) is considered, where lower and upper bounds for the flow are changed so that a given feasible flow becomes a maximum flow and the distance (considering l1 norm) between the initial vector of bounds and the modified vector is minimum. ..."
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Abstract: The general inverse maximum flow problem (denoted GIMF) is considered, where lower and upper bounds for the flow are changed so that a given feasible flow becomes a maximum flow and the distance (considering l1 norm) between the initial vector of bounds and the modified vector is minimum. Strongly and weakly polynomial algorithms for solving this problem are proposed. In the paper it is also proved that the inverse maximum flow problem where only the upper bound for the flow is changed (IMF) is a particular case of the GIMF problem.