Fast Vertex Guarding for Polygons

For a polygon P with n vertices, the vertex guarding problem asks for the minimum subset G of P ’s vertices such that every point in P is seen by at least one point in G. This problem is NP-complete and APX-hard. The first approximation algorithm (Ghosh, 1987) involves decomposing P into O ( n ) cells that are equivalence classes for visibility from the vertices of P . This discretized problem can then be treated as an instance of set cover and solved in O ( n ) time with a greedy O(logn)approximation algorithm. Ghosh (2010) recently revisited the algorithm, noting that minimum visibility decompositions for simple polygons (Bose et al., 2000) have only O ( n ) cells, improving the running time of the algorithm to O ( n ) for simple polygons. In this paper we show that, since minimum visibility decompositions for simple polygons have only O ( n ) cells of minimal visibility (Bose et al., 2000), the running time of the algorithm can be further improved to O ( n ) . This result was obtained independently by Jang and Kwon (2011). We extend the result of Bose et al. to polygons with holes, showing that a minimum visibility decomposition of a polygon with h holes has only O ( (h+ 1)n ) cells and only O ( (h+ 1)n ) cells of minimal visibility. We exploit this result to obtain a faster algorithm for vertex guarding polygons with holes. We then show that, in the same time complexity, we can attain approximation factors of O(log logopt) for simple polygons and O((1 + log (h+ 1)) logopt) for polygons with holes. ar X iv :1 10 1. 32 97 v2 [ cs .C G ] 1 6 Fe b 20 11