Why it matters

IP models many NP-hard problems. Understanding shapes optimization strategy.

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The architecture

Encode problem as IP: vars + constraints + objective.

Solve via branch-and-bound + cuts.

LP relaxation guides.

IP-based solvingInteger varsconstraints + objMILP solverB&B + cuts + heuristicsOptimal or boundwith proofCPLEX + Gurobi + SCIP + HiGHS handle huge industrial problems
IP hardness.
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How it works end to end

NP-hard in general.

LP relaxation: polynomial bound.

Modern solvers handle huge instances.