Engineering Art Galleries

Abstract

The Art Gallery Problem is one of the most well-known problems in Computational Geometry, with a rich history in the study of algorithms, complexity, and variants. Recently there has been a surge in experimental work on the problem. In this survey, we describe this work, show the chronology of developments, and compare current algorithms, including two unpublished versions, in an exhaustive experiment. Furthermore, we show what core algorithmic ingredients have led to recent successes.

Algorithms for art gallery illumination

“…Several versions of this problem were shown to be NP-hard, for example the problem restricted to guards placed on vertices of simple polygons, as shown by Lee and Lin [22]. See [12,23] for surveys on the AGP.…”

“…Currently, the state of the art is that optimal solutions for polygons with several thousand of vertices can be found efficiently. See [12] for a survey on that matter.…”

“…(1) Although both, the AGP with point guards and the AGP with vertex guards, are NPhard [22], many papers identified the vertex guard variant of the AGP to be much easier solvable in terms of practical algorithms than the point guard variant [4,7,12,20]. Since the AGPF is closely related to the AGP and our algorithms stem from our experience with the previous algorithms, see [12] for an overview, we start our investigation by tackling the discrete version. (2) The hardness in terms of tractability by practical algorithms of the AGPF lies in finding the darkest point in intermediate solutions (in this paper referred to as Primal Separation Problem (PSP)).…”

“…(2) The hardness in terms of tractability by practical algorithms of the AGPF lies in finding the darkest point in intermediate solutions (in this paper referred to as Primal Separation Problem (PSP)). Generating good/optimal guard locations has emerged to be the hardest part of the AGP with point guards [12]. So we refrain from entwining the PSP and guard generation to obtain meaningful insights on the tractability of the (novel) PSP.…”

“…Although A(G) and W are polynomial in size, they become very large from a practical point of view. Therefore, it is more efficient to solve this with the technique of iterative primal separation (compare [9,12,20] for the classical AGP), i.e., to leave out some constraints of the LP and start with a small set W , then solve AGPF τ (G, W ), and then check the A(G) for insufficiently lit points. If no such points exist, the solution is feasible.…”