The Parameterized Complexity of Guarding Almost Convex Polygons

Abstract

The Art Gallery problem is a fundamental visibility problem in Computational Geometry. The input consists of a simple polygon P, (possibly infinite) sets G and C of points within P, and an integer k; the task is to decide if at most k guards can be placed on points in G so that every point in C is visible to at least one guard. In the classic formulation of Art Gallery, G and C consist of all the points within P. Other well-known variants restrict G and C to consist either of all the points on the boundary of P or of all the vertices of P. Recently, three new important discoveries were made: the above mentioned variants of Art Gallery are all W[1]-hard with respect to k [Bonnet and Miltzow in 24th Annual European Symposium on Algorithms (Aarhus 2016)], the classic variant has an \({{\mathcal {O}}}(\log k)\)-approximation algorithm [Bonnet and Miltzow in 33rd International Symposium on Computational Geometry (Brisbane 2017)], and it may require irrational guards [Abrahamsen et al. in 33rd International Symposium on Computational Geometry (Brisbane 2017)]. Building upon the third result, the classic variant and the case where G consists only of all the points on the boundary of P were both shown to be \(\exists {\mathbb {R}}\)-complete [Abrahamsen et al. in 50th Annual ACM SIGACT Symposium on Theory of Computing (Los Angeles 2018)]. Even when both G and C consist only of all the points on the boundary of P, the problem is not known to be in NP. Given the first discovery, the following question was posed by Giannopoulos [Lorentz Workshop on Fixed-Parameter Computational Geometry (Leiden 2016)]: Is Art Gallery FPT with respect to r, the number of reflex vertices? In light of the developments above, we focus on the variant where G and C consist of all the vertices of P, called Vertex-Vertex Art Gallery. Apart from being a variant of Art Gallery, this case can also be viewed as the classic Dominating Set problem in the visibility graph of a polygon. In this article, we show that the answer to the question by Giannopoulos is positive: Vertex-Vertex Art Gallery is solvable in time \(r^{{{\mathcal {O}}}(r^2)}\hspace{0.55542pt}{\cdot }\hspace{1.66656pt}n^{{{\mathcal {O}}}(1)}\). Furthermore, our approach extends to assert that Vertex-Boundary Art Gallery and Boundary-Vertex Art Gallery are both FPT as well. To this end, we utilize structural properties of “almost convex polygons” to present a two-stage reduction from Vertex-Vertex Art Gallery to a new constraint satisfaction problem (whose solution is also provided in this paper) where constraints have arity 2 and involve monotone functions.