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 P oint G uard A rt G allery problem asks for a minimum set S such that every point in P is visible from a point in S . The V ertex G uard A rt G allery 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 ) n o ( 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 n O ( k ) algorithms that exist for both problems.
Highlights
Two points x, y in a simple polygon P are said to be visible to each other if the line segment between x and y is contained in P
The Vertex Guard Art Gallery problem asks for such a set S subset of the vertices of 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
Summary
To cite this version: Edouard Bonnet, Miltzow Tillmann. Parameterized Hardness of Art Gallery Problems. The Point Guard Art Gallery problem asks for a minimum set. 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. 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. ACM Reference Format: Édouard Bonnet and Tillmann Miltzow.
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