NP-Membership for the Boundary-Boundary Art-Gallery Problem

The boundary-boundary art-gallery problem asks, given a polygon P representing an art-gallery, for a minimal set of guards that can see the entire boundary of P (the wall of the art gallery), where the guards must be placed on the boundary. That is, for each point on the boundary, there should be a line segment connecting it to one of the guards that is contained in P . We show that this art-gallery variant is in NP, even if the polygon can have holes. In order to prove this, we develop a constraint-propagation procedure for continuous constraint satisfaction problems where each constraint involves at most 2 variables.

The X-Y variant of the art-gallery problem is the one where the guards must lie in X and need to see all of Y. Each of X and Y can be either the vertices of the polygon, the boundary of the polygon, or the entire polygon, giving 9 different variants. Previously, it was known that X-vertex and vertex-Y variants are all NP-complete and that the point-point, point-boundary, and boundary-point variants are ∃R-complete [Abrahamsen, Adamaszek, and Miltzow, JACM 2021][Stade, SoCG 2025]. However, the boundary-boundary variant was only known to lie somewhere between NP and ∃R.

The X-vertex and vertex-Y variants can be straightforwardly reduced to discrete set-cover instances. In contrast, we give example to show that a solution to an instance of the boundary-boundary art-gallery problem sometimes requires placing guards at irrational coordinates, so it unlikely that the problem can be easily discretized. Contents 5 Algorithms for M -2SAT 16 6 Irrational Coordinates 19 7 Acknowledgments 21 8 Conclusion 21