Partitioning orthogonal polygons into ≤ 8-vertex pieces, with application to an art gallery theorem

Abstract

We prove that every simply connected orthogonal polygon of n vertices can be partitioned into 3n+4 16 (simply connected) orthogonal polygons of at most 8 vertices. It yields a new and shorter proof of the theorem of A. Aggarwal that 3n+4 16 mobile guards are sufficient to control the interior of an n-vertex orthogonal polygon. Moreover, we strengthen this result by requiring combinatorial guards (visibility is only needed at the endpoints of patrols) and prohibiting intersecting patrols. This yields positive answers to two questions of O'Rourke [7, Section 3.4]. Our result is also a further example of the "metatheorem" that (orthogonal) art gallery theorems are based on partition theorems.

Mobile versus Point Guards

“…Kahn, Klawe, and Kleitman in 1980 [13], and a few years later Győri [10], and O'Rourke [18] proved that ⌊n/4⌋ point guards are sufficient and sometimes necessary to cover the interior of an orthogonal polygon of n vertices. Aggarwal proved in his thesis [1] that any n-vertex orthogonal polygon can be covered by at most ⌊ 3n+4 16 ⌋ mobile guards, and a strengthening of this result has been shown in [12]. These estimates are also shown to be sharp as extremal results.…”

“…There is, however, a polynomial time 3-approximation algorithm by Katz and Morgenstern [14] for the MSC problem for x-monotone orthogonal polygons without holes. Also, for an orthogonal polygon of n vertices, a covering set of mobile guards of cardinality at most ⌊(3n + 4)/16⌋ (which is the extremal bound shown by Aggarwal [1]) can be found in linear time [12]. In case holes are allowed, [2] give a polynomial time constant factor approximation algorithm.…”

“…In this paper, we study point and mobile guards that are equipped with rectangular vision, or r-vision for short: two points are visible to each other if their axis-parallel bounding rectangle is contained in the gallery. The results of [10,18,12] show that the worst case bounds on the number of point-and mobile guard required to control an n-vertex orthogonal polygon do not increase if line of sight vision is restricted to r-vision.…”