EADS talk by Mikkel Abrahamsen: The Art Gallery Problem is $\exists R$-complete
EADS talk by Mikkel Abrahamsen, PhD student at Department of Computer Science, University of Copenhagen (DIKU)
The Art Gallery Problem is $\exists R$-complete
We prove that the art gallery problem is equivalent under polynomial time reductions to deciding whether a system of polynomial equations over the real numbers has a solution. The art gallery problem is a classical problem in computational geometry, introduced in 1973 by Viktor Klee. Given a simple polygon P and an integer k, the goal is to decide if there exists a set G of k guards within P such that every point p in P is seen by at least one guard g in G. Each guard corresponds to a point in the polygon P, and we say that a guard g sees a point p if the line segment pg is contained in P.
The art gallery problem has stimulated a myriad of research in geometry and in algorithms. However, despite extensive research, the complexity status of the art gallery problem has not been resolved. It has long been known that the problem is NP-hard, but no one has been able to show that it lies in NP. Recently, the computational geometry community became more aware of the complexity class $\exists R$. The class $\exists R$ consists of problems that can be reduced in polynomial time to the problem of deciding whether a system of polynomial equations with integer coefficients and any number of real variables has a solution. It can be easily seen that $NP\subseteq \exists R$. We prove that the art gallery problem is $\exists R$-complete, implying that (1) any system of polynomial equations over the real numbers can be encoded as an instance of the art gallery problem, and (2) the art gallery problem is not in the complexity class NP unless $NP=\exists R$. As a corollary of our construction, we prove that for any real algebraic number $\alpha$ there is an instance of the art gallery problem where one of the coordinates of the guards equals $\alpha$ in any guard set of minimum cardinality. That rules out many geometric approaches to the problem.
Joint work with Anna Adamaszek and Tillmann Miltzow.