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32
Inferring Link Weights using EndtoEnd Measurements
 In ACM SIGCOMM Internet Measurement Workshop
, 2002
"... We describe a novel constraintbased approach to approximate ISP link weights using only endtoend measurements. Common routing protocols such as OSPF and ISIS choose leastcost paths using link weights, so inferred weights provide a simple, concise, and useful model of intradomain routing. Our ap ..."
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Cited by 108 (18 self)
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We describe a novel constraintbased approach to approximate ISP link weights using only endtoend measurements. Common routing protocols such as OSPF and ISIS choose leastcost paths using link weights, so inferred weights provide a simple, concise, and useful model of intradomain routing. Our approach extends routerlevel ISP maps, which include only connectivity, with link weights that are consistent with routing. Our inferred weights agree well with observed routing: while our inferred weights fully characterize the set of shortest paths between 8499% of the routerpairs, alternative models based on hop count and latency do so for only 4781% of the pairs.
Inverse optimization
 OPERATIONS RESEARCH
, 2001
"... In this paper, we study inverse optimization problems defined as follows. Let S denote the set of feasible solutions of an optimization problem P, let c be a specified cost vector, and x 0 be a given feasible solution. The solution x 0 may or may not be an optimal solution of P with respect to the c ..."
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Cited by 25 (2 self)
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In this paper, we study inverse optimization problems defined as follows. Let S denote the set of feasible solutions of an optimization problem P, let c be a specified cost vector, and x 0 be a given feasible solution. The solution x 0 may or may not be an optimal solution of P with respect to the cost vector c. The inverse optimization problem is to perturb the cost vector c to d so that x 0 is an optimal solution of P with respect to d and �d − c � p is minimum, where �d − c � p is some selected L p norm. In this paper, we consider the inverse linear programming problem under L 1 norm (where �d − c � p = ∑ i∈J w j�d j − c j�, with J denoting the index set of variables x j and w j denoting the weight of the variable j) and under L � norm (where �d −c � p = max j∈J �w j�d j −c j���. We prove the following results: (i) If the problem P is a linear programming problem, then its inverse problem under the L 1 as well as L � norm is also a linear programming problem. (ii) If the problem P is a shortest path, assignment or minimum cut problem, then its inverse problem under the L 1 norm and unit weights can be solved by solving a problem of the same kind. For the nonunit weight case, the inverse problem reduces to solving a minimum cost flow problem. (iii) If the problem P is a minimum cost flowproblem, then its inverse problem under the L 1 norm and unit weights reduces to solving a unitcapacity minimum cost flowproblem. For the nonunit weight case, the inverse problem reduces to solving a minimum cost flowproblem. (iv) If the problem P is a minimum cost flowproblem, then its inverse problem under the L � norm and unit weights reduces to solving a minimum mean cycle problem. For the nonunit weight case, the inverse problem reduces to solving a minimum costtotime ratio cycle problem. (v) If the problem P is polynomially solvable for linear cost functions, then inverse versions of P under the L 1 and L � norms are also polynomially solvable.
On the Use of an Inverse Shortest Paths Algorithm for Recovering Linearly Correlated Costs
, 1997
"... . This paper considers the inverse shortest paths problem where arc costs are subject to correlation constraints. The motivation for this research arises from applications in tra#c modelling and seismic tomography. A new method is proposed for solving this class of problems. It is constructed as ..."
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Cited by 17 (1 self)
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. This paper considers the inverse shortest paths problem where arc costs are subject to correlation constraints. The motivation for this research arises from applications in tra#c modelling and seismic tomography. A new method is proposed for solving this class of problems. It is constructed as a generalization of the algorithm presented in #1# for uncorrelated inverse shortest paths. Preliminary numerical experience with the new method is presented and discussed. yBelgian National Fund for Scienti#c Research Department of Mathematics Facult#es Universitaires ND de la Paix B5000 Namur, Belgium zDepartment of Mathematics Facult#es Universitaires ND de la Paix B5000 Namur, Belgium Keywords : graph theory, shortest paths, inverse problems, quadratic programming, tra#c modelling. On the use of an inverse shortest paths algorithm for recovering linearly correlated costs by D. Burtony and Ph.L. Tointz Report 91#09 June 10, 1997 Abstract. This paper considers the inver...
The inverse FermatWeber problem
, 2008
"... Given n points in the plane with nonnegative weights, the inverse FermatWeber problem consists in changing the weights at minimum cost such that a prespecified point in the plane becomes the Euclidean 1median. The cost is proportional to the increase or decrease of the corresponding weight. In cas ..."
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Cited by 14 (11 self)
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Given n points in the plane with nonnegative weights, the inverse FermatWeber problem consists in changing the weights at minimum cost such that a prespecified point in the plane becomes the Euclidean 1median. The cost is proportional to the increase or decrease of the corresponding weight. In case that the prespecified point does not coincide with one of the given n points, the inverse FermatWeber problem can be formulated as linear program. We derive a purely combinatorial algorithm which solves the inverse FermatWeber problem with unit cost in O(n log n) time. If the prespecified point coincides with one of the given n points, it is shown that the corresponding inverse problem can be written as convex problem and hence is solvable in polynomial time to any fixed precision. 1 Inverse and reverse location problems In recent years inverse and reverse optimization problems found an increased interest. In a reverse optimization problem, we are given a budget for modifying parameters of the problem. The goal is to modify parameters of the problem such that an objective function attains its best possible value subject to the given budget. The inverse optimization problem consists in changing parameters of the problem at minimum cost such that a prespecified solution becomes optimal. In one of the first papers on this subject, Burton and
Combinatorial Algorithms for Inverse Network Flow Problems
 WORKING PAPER, SLOAN SCHOOL OF MANAGEMENT, MIT
, 1998
"... An inverse optimization problem is defined as follows: Let S denote the set of feasible solutions of an optimization problem P, let c be a specified cost vector, and x 0 S. We want to perturb the cost vector c to d so that x 0 is an optimal solution of P with respect to the cost vector d, and  ..."
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Cited by 9 (7 self)
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An inverse optimization problem is defined as follows: Let S denote the set of feasible solutions of an optimization problem P, let c be a specified cost vector, and x 0 S. We want to perturb the cost vector c to d so that x 0 is an optimal solution of P with respect to the cost vector d, and d  c p is minimum, where . p denotes some selected L p norm. In this paper, we consider inverse minimum cut and minimum cost flow problems under the L 1 norm (where the objective is to minimize jJ w j d j  c j  for some index set J of variables), and under the L norm (where the objective is to minimize max{w j d j  c j : j J}). We show that the unit weight (that is, w j = 1 for all j J) inverse minimum cut problem under the L 1 norm reduces to solving a maximum flow problem, and under the L norm it requires solving a polynomial sequence of minimum cut problems. The unit weight inverse minimum cost flow problem under the L 1 norm reduces to solving a unit capacity minimum cost flow problem, and under the L norm it reduces to solving a minimum mean cycle problem. We also consider the nonunit weight versions of inverse minimum cut and minimum cost flow problems under the L norm.
The Inverse Shortest Paths Problem With Upper Bounds on Shortest Paths Costs
, 1997
"... We examine the computational complexity of the inverse shortest paths problem with upper bounds on shortest path costs, and prove that obtaining a globally optimum solution to this problem is NPcomplete. An algorithm for finding a locally optimum solution is proposed, discussed and tested. ..."
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Cited by 9 (1 self)
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We examine the computational complexity of the inverse shortest paths problem with upper bounds on shortest path costs, and prove that obtaining a globally optimum solution to this problem is NPcomplete. An algorithm for finding a locally optimum solution is proposed, discussed and tested.
The Complexity of an Inverse Shortest Paths Problem
 CONTEMPORARY TRENDS IN DISCRETE MATHEMATICS: FROM DIMACS AND DIMATIA TO THE FUTURE
, 1999
"... In this paper we study the complexity of an Inverse Shortest Paths Problem (ISPP). We show that the problem is intractable even in very restricted cases. In particular, we prove that the ISPP is NPcomplete in the planar case. Furthermore, we give a characterization of the class of graphs G d of giv ..."
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Cited by 6 (0 self)
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In this paper we study the complexity of an Inverse Shortest Paths Problem (ISPP). We show that the problem is intractable even in very restricted cases. In particular, we prove that the ISPP is NPcomplete in the planar case. Furthermore, we give a characterization of the class of graphs G d of given distances for which the ISPP is tractable: we give polynomial algorithms for special classes of G d and provide evidence that no polynomial time algorithms exist if the structure of G d is only slightly more complicated in a welldefined graphtheoretical sense. Finally, we discuss the situation for directed graphs.
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].
Shortest paths in stochastic networks with correlated link costs
 Computers and Mathematics With Applications
, 2005
"... Abstract. The objective is to minimize expected travel time from any origin to a specific destination in a congestible network with correlated link costs. Each link is assumed to be in one of two possible conditions. Conditional probability density functions for link travel times are assumed known f ..."
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Cited by 6 (0 self)
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Abstract. The objective is to minimize expected travel time from any origin to a specific destination in a congestible network with correlated link costs. Each link is assumed to be in one of two possible conditions. Conditional probability density functions for link travel times are assumed known for each condition. Conditions over the traversed links are taken into account for determining the optimal routing strategy for the remaining trip. This problem is treated as a multistage adaptive feedback control process. Each stage is described by the physical state (the location of the current decision point) and the information state (the service level of the previously traversed links). Proof of existence and uniqueness of the solution to the basic dynamic programming equations and a solution procedure are provided. Key Words. Shortest path, stochastic networks, dynamic programming, adaptive feedback control, correlated link costs.