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Karp's 21 NP-complete problems show that 0-1 integer linear programming is NP-hard. That is, an integer linear program with binary variables.

If we set the $c^T$ vector of the objective $\text {maximize } c^Tx$ to all one (unweighted, i.e., $c^T=(1,1,\dots,1)$) is the problem still NP-hard?

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We can encode satisfiability of a SAT instance as the feasibility of a 0-1 integer linear program. For feasibility, the objective function doesn't matter, so you can impose whatever constraint you wish on it.

For an example of how to express boolean or, boolean and, and boolean negation in a 0-1 integer linear program, see Express boolean logic operations in zero-one integer linear programming (ILP). This is all that's needed to express a SAT instance as a 0-1 integer linear program.

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  • $\begingroup$ Thanks. I was just reading Karp's proof and he only does feasibility so I guess that's what you saying too? $\endgroup$ – Mat Dec 30 '13 at 17:38
  • $\begingroup$ So if feasibility is NP-hard is maximization automatically NP-hard? $\endgroup$ – Mat Dec 30 '13 at 17:41
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    $\begingroup$ See if you can answer this question yourself - it will improve your understanding of NP-hardness. $\endgroup$ – Yuval Filmus Dec 30 '13 at 17:45
  • $\begingroup$ @Is the proof like: find the the maximum requires finding at least one feasible solution therefore if feasibility is NP-hard then maximization is NP-hard? $\endgroup$ – Mat Dec 30 '13 at 18:33
  • $\begingroup$ That's not a proof since your claim that finding a maximum required finding at least one feasible solution is unclear and could be false (depending on the interpretation). Try to use the definition of NP-hardness instead, perhaps by using a reduction. $\endgroup$ – Yuval Filmus Dec 30 '13 at 18:37

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