22 research outputs found
A (7/2)-Approximation Algorithm for Guarding Orthogonal Art Galleries with Sliding Cameras
Consider a sliding camera that travels back and forth along an orthogonal
line segment inside an orthogonal polygon with vertices. The camera
can see a point inside if and only if there exists a line segment
containing that crosses at a right angle and is completely contained in
. In the minimum sliding cameras (MSC) problem, the objective is to guard
with the minimum number of sliding cameras. In this paper, we give an
-time -approximation algorithm to the MSC problem on any
simple orthogonal polygon with vertices, answering a question posed by Katz
and Morgenstern (2011). To the best of our knowledge, this is the first
constant-factor approximation algorithm for this problem.Comment: 11 page
Partitioning orthogonal polygons into at most 8-vertex pieces, with application to an art gallery theorem
We prove that every simply connected orthogonal polygon of vertices can
be partitioned into (simply
connected) orthogonal polygons of at most 8 vertices. It yields a new and
shorter proof of the theorem of A. Aggarwal that mobile guards are sufficient to control the interior of
an -vertex orthogonal polygon. Moreover, we strengthen this result by
requiring combinatorial guards (visibility is only required at the endpoints of
patrols) and prohibiting intersecting patrols. This yields positive answers to
two questions of O'Rourke. Our result is also a further example of the
"metatheorem" that (orthogonal) art gallery theorems are based on partition
theorems.Comment: 20 pages, 12 figure
The Dispersive Art Gallery Problem
We introduce a new variant of the art gallery problem that comes from safety issues. In this variant we are not interested in guard sets of smallest cardinality, but in guard sets with largest possible distances between these guards. To the best of our knowledge, this variant has not been considered before. We call it the Dispersive Art Gallery Problem. In particular, in the dispersive art gallery problem we are given a polygon ? and a real number ?, and want to decide whether ? has a guard set such that every pair of guards in this set is at least a distance of ? apart.
In this paper, we study the vertex guard variant of this problem for the class of polyominoes. We consider rectangular visibility and distances as geodesics in the L?-metric. Our results are as follows. We give a (simple) thin polyomino such that every guard set has minimum pairwise distances of at most 3. On the positive side, we describe an algorithm that computes guard sets for simple polyominoes that match this upper bound, i.e., the algorithm constructs worst-case optimal solutions. We also study the computational complexity of computing guard sets that maximize the smallest distance between all pairs of guards within the guard sets. We prove that deciding whether there exists a guard set realizing a minimum pairwise distance for all pairs of guards of at least 5 in a given polyomino is NP-complete.
We were also able to find an optimal dynamic programming approach that computes a guard set that maximizes the minimum pairwise distance between guards in tree-shaped polyominoes, i.e., computes optimal solutions; due to space constraints, details can be found in the full version of our paper [Christian Rieck and Christian Scheffer, 2022]. Because the shapes constructed in the NP-hardness reduction are thin as well (but have holes), this result completes the case for thin polyominoes
Complexity of Chess Domination Problems
We study different domination problems of attacking and non-attacking rooks
and queens on polyominoes and polycubes of all dimensions. Our main result
proves that maximal domination is NP-complete for non-attacking queens and for
non-attacking rooks on polycubes of dimension three and higher. We also analyse
these problems for polyominoes and convex polyominoes, conjecture the
complexity classes and provide a computer tool for investigation. We have also
computed new values for classical queen domination problems on chessboards
(square polyominoes). For our computations, we have translated the problem into
an integer linear programming instance. Finally, using this computational
implementation and the game engine Godot, we have developed a video game of
minimal domination of queens and rooks on randomly generated polyominoes.Comment: 19 pages, 20 figures, 4 tables. Theorem 1 now for d>2, added results
on approximation, fixed typos, reorganised some proof
On -Guarding Thin Orthogonal Polygons
Guarding a polygon with few guards is an old and well-studied problem in
computational geometry. Here we consider the following variant: We assume that
the polygon is orthogonal and thin in some sense, and we consider a point
to guard a point if and only if the minimum axis-aligned rectangle spanned
by and is inside the polygon. A simple proof shows that this problem is
NP-hard on orthogonal polygons with holes, even if the polygon is thin. If
there are no holes, then a thin polygon becomes a tree polygon in the sense
that the so-called dual graph of the polygon is a tree. It was known that
finding the minimum set of -guards is polynomial for tree polygons, but the
run-time was . We show here that with a different approach
the running time becomes linear, answering a question posed by Biedl et al.
(SoCG 2011). Furthermore, the approach is much more general, allowing to
specify subsets of points to guard and guards to use, and it generalizes to
polygons with holes or thickness , becoming fixed-parameter tractable in
.Comment: 18 page