The Parameterized Complexity of Guarding Almost Convex Polygons

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 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 ∃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^𝒪(r^2)·n^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.