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Combinatorial algorithms for inverse network flow problems. (English) Zbl 1026.90089
Summary: 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 $$x^0\in S$$. 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 $$w\|d-c\|_p$$ is minimum, where $$\|\cdot \|_p$$ denotes some selected $$l_p$$ norm and $$w$$ is a vector of weights. In this paper, we consider inverse minimum-cut and minimum-cost flow problems under the $$l_1$$ normal (where the objective is to minimize $$\sum_{j\in J}w_j |d_j -c_j|$$ for some index set $$J$$ of variables) and under the $$l_\infty$$ norm (where the objective is to minimize $$\max\{w_j |d_j-c_j |:j \in J \})$$. We show that the unit weight (i.e., $$w_j=1$$ for all $$j\in J)$$ inverse minimum-cut problem under the $$l_1$$ norm reduces to solving a maximum-flow problem, and under the $$l_\infty$$ 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 circulation problem, and under the $$l_\infty$$ 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 $$l_\infty$$ norm.

##### MSC:
 90C35 Programming involving graphs or networks 90B10 Deterministic network models in operations research
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