DeepAI
Log In Sign Up

The Parameterized Complexity of Guarding Almost Convex Polygons

03/17/2020
by   Akanksha Agrawal, et al.
0

Art Gallery 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, ESA'16], the classic variant has an O(log k)-approximation algorithm [Bonnet and Miltzow, SoCG'17], and it may require irrational guards [Abrahamsen et al., SoCG'17]. 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., STOC'18]. 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 Center Workshop, 2016]: Is Art Gallery FPT with respect to r, the number of reflex vertices? We focus on the variant where G and C are all the vertices of P, called Vertex-Vertex Art Gallery. We show that Vertex-Vertex Art Gallery is solvable in time r^O(r^2)n^O(1). Our approach also extends to assert that Vertex-Boundary Art Gallery and Boundary-Vertex Art Gallery are both FPT. 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.

READ FULL TEXT

page 1

page 2

page 3

page 4

07/02/2019

A Constant-Factor Approximation Algorithm for Vertex Guarding a WV-Polygon

The problem of vertex guarding a simple polygon was first studied by Sub...
09/19/2019

A note on minimal art galleries

We will consider some extensions of the polygonal art gallery problem. I...
06/28/2017

On Compatible Triangulations with a Minimum Number of Steiner Points

Two vertex-labelled polygons are compatible if they have the same clockw...
03/02/2018

An improved FPT algorithm for Independent Feedback Vertex Set

We study the Independent Feedback Vertex Set problem - a variant of the ...
11/17/2021

The Polygon Burning Problem

Motivated by the k-center problem in location analysis, we consider the ...
06/24/2018

On r-Simple k-Path and Related Problems Parameterized by k/r

Abasi et al. (2014) and Gabizon et al. (2015) studied the following prob...
05/10/2022

Burning Number for the Points in the Plane

The burning process on a graph G starts with a single burnt vertex, and ...