Combinatorial algorithms for inverse network flow problems

An inverse optimization problems 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 xO be a given feasible solution. We want to perturb the cost vector c to d so that x is an optimal solution of P with respect to the cost vector d , and lid clip is minimum, where 11.llp denotes some selected Lp norm. In this paper, we consider inverse versions of the following network flow problems: the shortest path problem, the assignment problem, the minimum cut problem, and the minimum cost flow problem. We consider inverse problems under the L1 norm (where the objective is to minimize jJ wjldj cjl), and under the Lo norm (where the objective is to minimize max{wjldj cjl : j J}). We show that the inverse version of each of the problem considered under the L 1 norm reduces to solving a problem for the same kind; that is, an inverse shortest path problem reduces to a shortest path problem, an inverse assignment problem reduces to an assignment problem, and so on. We next show that inverse versions of the shortest path problem, the assignment problem, and the minimum cost flow problem, under the Lo norm reduce to solving a minimum mean cycle problem (where we wish to identify a cycle whose cost divided by the number of arcs in it is minimum). We also consider the inverse minimum cut problem under the Lo norm and suggest a polynomial-time binary search algorithm. 1 Sloan School of Management, MIT, Cambridge, MA 02139, USA; On leave from Indian Institute of Technology, Kanpur 208 016, INDIA. 2 Sloan School of Management, MIT, Cambridge, MA 02139, USA.