General restricted inverse assignment problems under $$l_1$$ and $$l_\infty $$ norms

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In this paper, we study the general restricted inverse assignment problems, in which we can only change the costs of some specific edges of an assignment problem as less as possible, so that a given assignment becomes the optimal one. Under $$l_1$$ norm, we formulate this problem as a linear programming. Then we mainly consider two cases. For the case when the specific edges are only belong to the given assignment, we show that this problem can be reduced to some variations of the minimum cost flow problems. For the case when every specific edge does not belong to the given assignment, we show that this problem can be solved by a minimum cost circulation problem. In both cases, we present some combinatorial algorithms which are strongly polynomial. We also study this problem under the $$l_\infty $$ norm. We propose a binary search algorithm and prove that the optimal solution can be obtained in polynomial time.