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.

翻译：设$G=(V, E)$为一个图，且$G$的每个顶点均配备一个灯和一个按钮。每个按钮可为$\sigma^+$型或$\sigma$型。假设初始状态下部分灯亮、部分灯灭。若按下顶点$x$处的按钮会改变$x$及其在$G$中邻点的灯状态（或仅改变$x$邻点的灯状态），则该按钮为$\sigma^+$型（相应地，$\sigma$型）。假设存在子集$X\subseteq V$，使得按下$X$中顶点处的按钮可点亮$G$中所有顶点的灯。特别地，已知当初始所有灯均熄灭且所有按钮均为$\sigma^+$型时该条件成立。即使初始所有灯均熄灭且所有按钮均为$\sigma^+$型，寻找最小规模的此类集合$X$是NP困难的。本文采用线性代数方法，为该问题设计了一个多项式时间近似算法，使得对于算法构造的集合$X$，有$|X|\le \min\{r,(|V|+{\rm opt})/2\}$，其中$r$为$G$的（修正）邻接矩阵的秩，${\rm opt}$为该问题最优解的大小。据我们所知，这是该问题首个具有非平凡近似保证的多项式时间近似算法。