Parameterized Hardness of Art Gallery Problems

Abstract

Given a simple polygon P on n vertices, two points x, y in P are said to be visible to each other if the line segment between x and y is contained in P. The Point Guard Art Gallery problem asks for a minimum set S such that every point in P is visible from a point in S. The Vertex Guard Art Gallery problem asks for such a set S subset of the vertices of P. A point in the set S is referred to as a guard. For both variants, we rule out any f(k)no(k / log k) algorithm, where k := |S| is the number of guards, for any computable function f, unless the exponential time hypothesis fails. These lower bounds almost match the nO(k) algorithms that exist for both problems.