We consider the Minimum Convex Partition problem: Given a set P of n points in the plane, draw a plane graph G on P, with positive minimum degree, such that G partitions the convex hull of P into a minimum number of convex faces. We show that Minimum Convex Partition is NP-hard, and we give several approximation algorithms, from an O(log OPT)-approximation running in O(n^8)-time, where OPT denotes the minimum number of convex faces needed, to an O(sqrt(n) log n)-approximation algorithm running in O(n^2)-time. We say that a point set is k-directed if the (straight) lines containing at least three points have up to k directions. We present an O(k)-approximation algorithm running in n^O(k)-time. Those hardness and approximation results also holds for the Minimum Convex Tiling problem, defined similarly but allowing the use of Steiner points. The approximation results are obtained by relating the problem to the Covering Points with Non-Crossing Segments problem. We show that this problem is NP-hard, and present an FPT algorithm. This allows us to obtain a constant-approximation FPT algorithm for the Minimum Convex Partition Problem where the parameter is the number of faces.

翻译：我们考虑最小 Convex 分区问题 : 如果在平面上设置了n点的固定点, 请在 P上绘制一个平面图G, 以最小度为正数, 使 G 将 P 的 convex 柱体分隔成最小数的 convex 面孔。 我们显示最小 Convex 分区是 NP- 硬化的, 我们从O( n) 8 时间运行的 O( OL OOP)- approcolation 算法中提供几种近似算法, o( log On) 表示需要最小的 convex 面孔数, 到 O( n) 时的 O( sqrt (n) log n) 接近率算法。 我们说, 如果至少包含 3 个点的( straight) 直线达到 kdroad 方向, 则设定一个点为 kpoint 。 我们给出了 O( k)- a adroc) ad 算算算算法的最小值和最起码的 Rassional 问题 。 解算法 显示当前 问题, 解算法 问题是非 问题 。