Approximation Algorithm of Minimum All-Ones Problem for Arbitrary Graphs
Chen Wang, Chao Wang, Gregory Z. Gutin, Xiaoyan Zhang
TL;DR
This work studies the min generalized all-ones problem on arbitrary graphs, where lamps and switches (σ^+-type or σ-type) interact according to graph adjacency. It introduces a linear-algebraic formulation over GF(2) using a modified adjacency matrix and an initial state, and derives a fundamental solution set that describes all feasible solutions when one exists. The authors present a polynomial-time approximation algorithm that computes a solution X with |X| ≤ min{r, (|V|+opt)/2}, where r is the rank of the modified adjacency matrix and opt is the optimal size, achieving sol ≤ r and sol ≤ (n+opt)/2 with O(n^3) time (reducible to O(n(n−r)) under certain form conditions). This yields the first nontrivial approximation guarantee for min generalized all-ones on general graphs, with implications for related all-ones variants and potential future improvements toward constant-factor guarantees. The work further identifies open questions about constant-factor approximations and extensions to the all-colors problem.
Abstract
Let $G=(V, E)$ be a graph and let each vertex of $G$ has a lamp and a button. Each button can be of $σ^+$-type or $σ$-type. Assume that initially some lamps are on and others are off. The button on vertex $x$ is of $σ^+$-type ($σ$-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 $σ^+$-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 $σ^+$-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.