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Integer programming is said to be NP-complete. However, I think formulating a problem into ILP can't prove the problem to be NP-hard. Is there any example of problem that can be modeled into ILP but has a polynomial time?

Peter Chan
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    Maximum flow seems to fit the bill? – David Eisenstat Jan 26 '22 at 14:12
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    Find the maximum of a list of numbers? But formulating a problem as an ILP shows it's NP, not NP hard. – Paul Hankin Jan 26 '22 at 14:31
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    There's a confusion here. "Formulating a problem as a particular instance of ILP" means you've shown that your problem is **not harder than** ILP. Just because we don't have an algorithm to solve **all ILP problems** in polynomial time, doesn't mean that **your particular problem** cannot be solved in polynomial time. – Stef Jan 26 '22 at 15:48
  • (I apologize for the triple-negation in my previous comment) – Stef Jan 26 '22 at 15:49
  • Sorry I just found out I have a typo. I meant I think formulating a problem into ILP can't prove the problem to be NP-hard. – Peter Chan Jan 28 '22 at 14:58
  • @DavidEisenstat Could you explain more? I am not familiar with it. – Peter Chan Jan 28 '22 at 15:00
  • [Maximum flow](https://en.wikipedia.org/wiki/Maximum_flow_problem) is one of the harder-looking "easy" (i.e., in P) problems that you can model with ILP. – David Eisenstat Jan 28 '22 at 15:11

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