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Given a 0-1 matrix $A$, is there an efficient way to find all 0-1 vectors $x$ such that $Ax = v$ where the entries of $v$ belong to a set $\{a,b\} \subseteq \mathbb{Z}$ of size $2$? Note that $v$ is not a fixed vector, it can vary over all $2^n$ vectors where $n$ is the dimension.

Of course, I require something which would be more efficient than going through all the $2^n$ possible values for $v$ where $n$ is the length of vector $v$ and doing Gaussian elimination each time. Is there something more efficient than that?

Edit: The motivation comes from the following problem. Given a $k$-uniform family $F$ of subsets of $[n] := \{1,\ldots,n\}$ determine all subsets of $[n]$ that intersect each member of $F$ in $1$ or $k$ points. For $k = 3$ this can be done easily by computing the nullspace of the incidence matrix over $GF(2)$ since a subset intersects a member of $F$ in $1$ or $3$ points if and only if its complement intersects that member in $0$ or $2$ points. I was wondering if there are similar approaches for other values of $k$. It doesn't look like the trick generalises to larger $k$.

Olexandr Konovalov
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Anurag
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  • Are the values which entries of $v$ can take predetermined? – 5xum May 09 '14 at 09:36
  • Yes. There are fixed integers a and b such that the entries of v are either a or b. – Anurag May 09 '14 at 10:28
  • You could use Gaussian elimination to solve the system - see this excerpt of Montanari's book "Information, Physics and Computation" (http://www.stanford.edu/~montanar/RESEARCH/BOOK/partE.pdf). – Batman May 09 '14 at 11:45
  • Yes, but I don't want go through all 2^n possibilities of v. The vector v on the right side is not fixed here. We just know that v can be any of the 2^n vectors.

    Moreover, the techniques discussed there are for solving the equation over Z_2, which is not the case here.

     – Anurag May 09 '14 at 12:10
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    Constraint programming solvers, e.g. Minion may help with this. – Olexandr Konovalov Oct 15 '14 at 19:29

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