Combinatorial algorithms for inverse network flow problems

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AbstractAn 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 x0 ∈ S. We want to perturb the cost vector c to d so that x0 is an optimal solution of P with respect to the cost vector d, and w∥d − c∥p is minimum, where ∥ · ∥p denotes some selected lp norm and w is a vector of weights. In this paper, we consider inverse minimum‐cut and minimum‐cost flow problems under the l1 normal (where the objective is to minimize ∑j∈Jwj|dj − cj| for some index set J of variables) and under the l∞ norm (where the objective is to minimize max{wj|dj − cj|: j ∈ J}). We show that the unit weight (i.e., wj = 1 for all j ∈ J) inverse minimum‐cut problem under the l1 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 l1 norm reduces to solving a unit capacity minimum‐cost circulation 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. © 2002 Wiley Periodicals, Inc.