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Equivalently, you want to remove the smallest number of constraints so that the resulting problem is feasible. You can do this implicitly, without enumerating all Irreducible Infeasible Sets, by solving an auxiliary ILP problem. Introduce a binary variable $z_i$ for each constraint (including variable bounds as a special case) $$\sum_j a_{i,j} x_j \le b_i$$...

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