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Since a MILP problem is not necessarily NP hard. How could I demonstrate that a MILP problem is actually NP hard? There exists some smart and easy method to do that? Many thanks!

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    $\begingroup$ This, like for most problems in NP, depends on the problem and can be done with a reduction. Unless you specify the class of MILP problems you wish to show are NP-hard, we cannot really give a more specific answer. However, this may be related: cs.stackexchange.com/questions/89140/… $\endgroup$
    – Discrete lizard
    Apr 26, 2019 at 14:56

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In general, a general MILP was proved to be NP-Hard. This is a well-known proof and can be easily found in any textbook (or even on Wikipedia).
However, you can see one specific MILP problem as an 'instance' of the general MILP problem. Thus, you still need to show its complexity.


In fact, several MILP problems are NP-Hard:

  1. Elementary Shortest Path (if the graph has negative cycles)
  2. Degree-constrained Minimum Spanning Tree
  3. Vehicle Routing Problem

However, many others are shown to be solvable in polynomial-time:

  1. Job scheduling
  2. Minimum Spanning Tree
  3. Aircraft scheduling problem

You can demonstrate that a MILP problem is NP-Hard in the same way you do with any other optimization problem.

All you need to do is to reduce an NP-Hard optimization problem $\sigma$ to your MILP, construct a solution for MILP, and shows a polynomial-time transformation of your obtained solution to a solution of $\sigma$.

Sadly, there is no standard technique or proof method that can be applied to show the complexity of any MILP problems and you need to use your creativity (and mathematical skills, in many cases) to proof the complexity of your problem.

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