Consider the following system of linear equations modulo 2:

$A.X + B.Y = Z, $

where $A$ is a non-singular(modulo 2) $n$ x $n$ boolean matrix, $B $ is $n$ x $m$ boolean matrix, $X$ is n-dimensional boolean(unknown) vector, $Y$ is $m$-dimensional boolean(unknown) vector and $Z$ is $n$-dimensional boolean unknown vector.

So, solutions of the equation above are pairs of boolean vectors $(X,Y)$.

Prove that the number of solutions is equal $2^{m}$. Use this observation to get a polynomial time algorithm to count a number of 2-colorings.

I see that the products are defined. We would end up with $Z$ as a (nx1) vector that I assume is mod2 as well. All possible combinations of this boolean solution is $2^{n}$. I don't know why the question state it as $m$. Apparently, I need to use the Gauss elimination algorithm for 2-colorability in order to solve this problem.

Any ideas? Thank you.


Since $A$ is non-singular the equation is equivalent to $X = A^{-1}(Z-BY)$. So, for any choice of $Y$ you get a solution to the system. So, the number of solutions is the same as the number of possible $Y$. That is $2^m$.

  • $\begingroup$ Thank you. It's a very nice observation and shows my rusty Algebra. However, I am still confused about, "Use this observation to get a polynomial time algorithm to count a number of 2-colorings." I guess that the connection between a 2-colorability graph problem and the matrix is because of mod2 constriction? I know that the 2-coloring problem reduces to a decidable problem iff the graph is bi-partite or not; this takes polynomial time. ' $\endgroup$ – knowKnothing Dec 18 '13 at 11:49
  • $\begingroup$ What is the precise statement of "count a number of two coloring" problem? $\endgroup$ – hbm Dec 18 '13 at 12:59
  • $\begingroup$ Yes, I don't understand the statement either. I can only guess what the problem means, I could state some algorithm for decidability of a bi-partite graph. But, I am not fully sure what the "count a number of two coloring" statement means. $\endgroup$ – knowKnothing Dec 18 '13 at 13:23
  • $\begingroup$ I think that the statement refers to specific colorability and linear algebra differences. For Linear Algebra, vector [1 0 0] is different than vector [0 1 0], but for graph-colorability theory both vectors represent 3 colorability valid vectors. So, I think the question of counting a number of 2-colorings refer to find values of Y that are valid for colorability. $\endgroup$ – knowKnothing Dec 19 '13 at 20:35
  • $\begingroup$ What kind of 2-coloring are you considering? If you are considering proper vertex 2-coloring, then any connected bipartite graph has exactly 2 proper colorings. However, if you are considering any 2-coloring, them the number is $2^k$ where $k$ is the number of vertices of the graph. $\endgroup$ – hbm Dec 19 '13 at 20:41

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