Performance guarantees on a sweep-line heuristic for covering rectilinear polygons with rectangles
SIAM Journal on Discrete Mathematics
Journal of Algorithms
Approximation algorithms for NP-hard problems
Approximation algorithms for NP-hard problems
Covering rectilinear polygons with axis-parallel rectangles
STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
The Power of Non-Rectilinear Holes
Proceedings of the 9th Colloquium on Automata, Languages and Programming
Inapproximability Results for Guarding Polygons without Holes
ISAAC '98 Proceedings of the 9th International Symposium on Algorithms and Computation
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The problem Minimum Convex Cover of covering a given polygon with a minimum number of (possibly overlapping) convex polygons is known to be NP-hard, even for polygons without holes [3]. We propose a polynomial-time approximation algorithm for this problem for polygons with or without holes that achieves an approximation ratio of O(log n), where n is the number of vertices in the input polygon. To obtain this result, we first show that an optimum solution of a restricted version of this problem, where the vertices of the convex polygons may only lie on a certain grid, contains at most three times as many convex polygons as the optimum solution of the unrestricted problem. As a second step, we use dynamic programming to obtain a convex polygon which is maximum with respect to the number of "basic triangles" that are not yet covered by another convex polygon.We obtain a solution that is at most a logarithmic factor off the optimum by iteratively applying our dynamic programming algorithm. Furthermore, we show that Minimum Convex Cover is APX-hard, i.e., there exists a constant δ 0 such that no polynomial-time algorithm can achieve an approximation ratio of 1 + δ. We obtain this result by analyzing and slightly modifying an already existing reduction [3].