Abstract
In this paper, we study the inverse problem of submodular functions on digraphs. Given a feasible solution x* for a linear program generated by a submodular function defined on digraphs, we try to modify the coefficient vector c of the objective function, optimally and within bounds, such that x* becomes an optimal solution of the linear program. It is shown that the problem can be formulated as a combinatorial linear program and can be transformed further into a minimum cost circulation problem. Hence, it can be solved in strongly polynomial time. We also give a necessary and sufficient condition for the feasibility of the problem. Finally, we extend the discussion to the version of the inverse problem with multiple feasible solutions.
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Cai, M., Yang, X. & Li, Y. Inverse Problems of Submodular Functions on Digraphs. Journal of Optimization Theory and Applications 104, 559–575 (2000). https://doi.org/10.1023/A:1004685508517
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DOI: https://doi.org/10.1023/A:1004685508517