An Approximation Algorithm for Minimum Convex Cover with Logarithmic Performance Guarantee

Abstract

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].KeywordsApproximation AlgorithmApproximation RatioConvex PolygonClockwise OrderBasic TriangleThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.