  Let $G=(V, E)$ be a graph and let each vertex of $G$ has a lamp and a button.
Each button can be of $\sigma^+$-type or $\sigma$-type.
  Assume that initially some lamps are on and others are off. The button on
vertex $x$ is of $\sigma^+$-type ($\sigma$-type, respectively) if pressing the
button changes the lamp states on $x$ and on its neighbors in $G$ (the lamp
states on the neighbors of $x$ only, respectively). Assume that there is a set
$X\subseteq V$ such that pressing buttons on vertices of $X$ lights all lamps
on vertices of $G$. In particular, it is known to hold when initially all lamps
are off and all buttons are of $\sigma^+$-type.
  Finding such a set $X$ of the smallest size is NP-hard even if initially all
lamps are off and all buttons are of $\sigma^+$-type. Using a linear algebraic
approach we design a polynomial-time approximation algorithm for the problem
such that for the set $X$ constructed by the algorithm, we have $|X|\le
\min\{r,(|V|+{\rm opt})/2\},$ where $r$ is the rank of a (modified) adjacent
matrix of $G$ and ${\rm opt}$ is the size of an optimal solution to the
problem.
  To the best of our knowledge, this is the first polynomial-time approximation
algorithm for the problem with a nontrivial approximation guarantee.
