Restless Guards: Robust Guarding of Polyominoes

Abstract

The art gallery problem is a fundamental problem whose practical applications have inspired numerous variants. In this work, we investigate a setting in which full coverage of the gallery shall be ensured despite locational uncertainty of the guards. We study this problem for polyominoes under rectangular visibility, where a $k$-restless guard assigned to a cell may move to any cell within distance at most $k$. While all polyominoes are guardable for $k\in\{0, 1\}$, the same is not true for all $k\geq 2$. However, for a given polyomino $P$ and integer $k$, we present an algorithm that identifies a $k$-restless guard set of $P$ in polynomial time, or decides that no such set exists. For the cases $k\in\{0,1\}$, where $k$-restless guard sets always exist, we provide tight worst-case bounds on the number of $k$-restless guards that are always sufficient and sometimes necessary to guard a hole-free polyomino with $n$ vertices or $m$ cells. Moreover, we show that computing a minimum guard set is NP-hard. Complementing the hardness, we provide polynomial-time algorithms for solving the problem optimally for histograms, and leverage this to obtain a $20k$-approximation for polyominoes without holes.