Expanding Visibility Polygons by Mirrors upto at least K units

Abstract

We consider extending visibility polygon $(VP)$ of a given point $q$ $(VP(q))$, inside a simple polygon $\P$ by converting some edges of $\P$ to mirrors. We will show that several variations of the problem of finding mirror-edges to add at least $k$ units of area to $VP(q)$ are NP-complete, or NP-hard. Which $k$ is a given value. We deal with both single and multiple reflecting mirrors, and also specular or diffuse types of reflections. In specular reflection, a single incoming direction is reflected into a single outgoing direction. In this paper diffuse reflection is regarded as reflecting lights at all possible angles from a given surface. The paper deals with finding mirror-edges to add \emph{at least} $k$ units of area to $VP(q)$. In the case of specular type of reflections we only consider single reflections, and the multiple case is still open. Specular case of the problem is more tricky. We construct a simple polygon for every given instance of a 3-SAT problem. There are some specific spikes which are visible only by some particular mirror-edges. Consequently, to have minimum number of mirror-edges it is required to choose only one of these mirrors to see a particular spike. There is a reduction polygon which contains a clause-gadget corresponding to every clause, and a variable-gadget corresponding to every variable. 3-SAT formula has $n$ variables and $m$ clauses, so the minimum number of mirrors required to add an area of at least $k$ to $V P(q)$ is $l = 3m+n+1$ if and only if the 3-SAT formula is satisfiable. This reduction works in these two cases: adding at least $k$ vertex of $\P$ to $VP(q)$, and expanding $VP(q)$ at least $k$ units of area.