Approximation Algorithm of Minimum All-Ones Problem for Arbitrary Graphs

TL;DR

This paper introduces the first polynomial-time approximation algorithm for the NP-hard minimum all-ones problem on arbitrary graphs, providing a nontrivial approximation guarantee based on graph matrix rank.

Contribution

It presents a novel polynomial-time approximation algorithm with a proven approximation bound for the minimum all-ones problem, expanding computational tools for this class of problems.

Findings

The algorithm guarantees a solution size within a factor related to the graph's matrix rank.

It achieves an approximation ratio of at most half the sum of the number of vertices and the optimal solution.

This is the first known polynomial-time approximation with a nontrivial guarantee for this 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 $\sigma^+$-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.