An Upper Bound on the k-Modem Illumination Problem

A variation on the classical polygon illumination problem was introduced in [Aichholzer et al. EuroCG’09]. In this variant light sources are replaced by wireless devices called [Formula: see text]-modems, which can penetrate a fixed number [Formula: see text], of “walls”. A point in the interior of a polygon is “illuminated” by a [Formula: see text]-modem if the line segment joining them intersects at most [Formula: see text] edges of the polygon. It is easy to construct polygons of [Formula: see text] vertices where the number of [Formula: see text]-modems required to illuminate all interior points is [Formula: see text]. However, no non-trivial upper bound is known. In this paper we prove that the number of kmodems required to illuminate any polygon of [Formula: see text] vertices is [Formula: see text]. For the cases of illuminating an orthogonal polygon or a set of disjoint orthogonal segments, we give a tighter bound of [Formula: see text]. Moreover, we present an [Formula: see text] time algorithm to achieve this bound.

COMMON DEVELOPMENTS OF THREE INCONGRUENT ORTHOGONAL BOXES

We investigate common developments that can fold into several incongruent orthogonal boxes. It was shown that there are infinitely many orthogonal polygons that fold into two incongruent orthogonal boxes in 2008. In 2011, it was shown that there exists an orthogonal polygon that folds into three boxes of size 1 × 1 × 5, 1 × 2 × 3, and 0 × 1 × 11. However it remained open whether there exists an orthogonal polygon that folds into three boxes of positive volume. We give an affirmative answer to this open problem. We show how to construct an infinite number of orthogonal polygons that fold into three incongruent orthogonal boxes.

LINEAR-TIME 3-APPROXIMATION ALGORITHM FOR THE r-STAR COVERING PROBLEM

The complexity status of the minimum r-star cover problem for orthogonal polygons had been open for many years, until 2004 when Ch. Worman and J. M. Keil proved it to be polynomially tractable (Polygon decomposition and the orthogonal art gallery problem, IJCGA 17(2) (2007), 105-138). However, since their algorithm has Õ(n17)-time complexity, where Õ(·) hides a polylogarithmic factor, and thus it is not practical, in this paper we present a linear-time 3-approximation algorithm. Our approach is based upon the novel partition of an orthogonal polygon into so-called o-star-shaped orthogonal polygons.