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I'm trying to determine whether it is possible to decide if a system of $n$ linear equations with integer coefficients and $n$ variables has a solution in $\{0, 1\}^n$ in polynomial time.

Additionally, all of the coefficients of $A$ are in $\{-1, 0, 1\}$, but I couldn't find a way to use that.

The trivial case is if the (matrix $A$ of the) system is invertible, there is only one solution, it is easy to check whether all of them belong to $\{0,1\}$.

However, if you have infinitely many solutions, and $k$ free variable, I can't find a way to do better than check all the $2^k$ possibilities.

  • Do you know any algorithm to do so in polynomial time ?

I also tried to do a reduction from SAT (or some variant with n clauses and n variables in each clause, to show that it is NP complete), but because of the fact that we have $Ax = b$ and not $Ax \geq b$, I couldn't do that either.

  • Do you have a reduction to show that this problem is NP complete ?
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    $\begingroup$ Try reducing from 1-IN-3SAT. $\endgroup$ – Yuval Filmus Dec 14 '18 at 19:41
  • $\begingroup$ @YuvalFilmus that is an answer. And my algorithm must be wrong. But why? $\endgroup$ – Solomonoff's Secret Dec 14 '18 at 19:49
  • $\begingroup$ Your algorithm ignores the restriction that all variables be $\{0,1\}$. $\endgroup$ – Yuval Filmus Dec 14 '18 at 19:52
  • $\begingroup$ @Solomonoff'sSecret The problem comes with free variables in the system. $\endgroup$ – GBat Dec 14 '18 at 20:22
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1-IN-3SAT is the following problem: given a collection of 3-clauses (just like 3SAT), decide whether there is a truth assignment which satisfies exactly one literal in each clause. 1-IN-3SAT is known to be NP-complete, see for example here. You can reduce 1-IN-3SAT to your problem – I'll let you work out the details.

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