Facets for Art Gallery Problems

Abstract

We demonstrate how polyhedral methods of mathematical programming can be developed for and applied to computing optimal solutions for large instances of a classical geometric optimization problem with an uncountable number of constraints and variables.The Art Gallery Problem (AGP) asks for placing a minimum number of stationary guards in a polygonal region P, such that all points in P are guarded. The AGP is NP-hard, even to approximate. Due to the infinite number of points to be guarded as well as possible guard positions, applying mathematical programming methods for computing provably optimal solutions is far from straightforward.In this paper, we use an iterative primal-dual relaxation approach for solving AGP instances to optimality. At each stage, a pair of LP relaxations for a finite candidate subset of primal covering and dual packing constraints and variables is considered; these correspond to possible guard positions and points that are to be guarded.Of particular interest are additional cutting planes for eliminating fractional solutions. We identify two classes of facets, based on Edge Cover and Set Cover (SC) inequalities. Solving the separation problem for the latter is NP-complete, but exploiting the underlying geometric structure of the AGP, we show that large subclasses of fractional SC solutions cannot occur for the AGP. This allows us to separate the relevant subset of facets in polynomial time.Finally, we characterize all facets for finite AGP relaxations with coefficients in {0, 1, 2}. We demonstrate the practical usefulness of our approach with improved solution quality and speed for a wide array of large benchmark instances.KeywordsArt Gallery Problemgeometric optimizationalgorithm engineeringset cover polytopesolving NP-hard problem instances to optimality