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The minimum cost flow problem with conflicts is a recent extension of the ordinary minimum cost flow problem. It includes flow compatibility restrictions in addition to flow balance equalities and capacity constraints: at most one of the conflicting arcs can have positive flow. In this work we study this extension of the minimum cost flow problem, which can be faced in many real‐world applications. We give mixed‐integer programming formulations of the problem after briefly analyzing its complexity and propose two exact solution algorithms. One of them is a branch‐and‐bound procedure with combinatorial branching rules based on conflicts in an optimal solution of the relaxations. The other one is a Russian Doll Search method exploring the maximal stable sets of the conflict graph, which is obtained from the original network with the purpose of representing the conflict relations among arcs. Also, a set of preprocessing procedures is introduced with the aim of reducing the size of the problem. We can say that the new algorithms are very efficient according to the results of extensive computational experiments realized on a large set of test problems.

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