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Computer Science, Computational Geometry, cs.CG,Computer Science, Data Structures and Algorithms, cs.DS,Mathematics, Optimization and Control, math.OC
Abstract:
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
problem is known to be NP-hard, and its inherent continuous structure (with
both the set of points that need to be guarded and the set of points that can
be used for guarding being uncountably infinite) makes it difficult to apply a
straightforward formulation as an Integer Linear Program. 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.
Particularly useful are 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, 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. We also characterize all
facets for finite AGP relaxations with coefficients in {0, 1, 2}.
Finally, we demonstrate the practical usefulness of our approach. Our cutting
plane technique yields a significant improvement in terms of speed and solution
quality due to considerably reduced integrality gaps as compared to the
approach by Kr\"oller et al.
