In the maximum independent set of convex polygons problem, we are given a set of $n$ convex polygons in the plane with the objective of selecting a maximum cardinality subset of non-overlapping polygons. Here we study a special case of the problem where the edges of the polygons can take at most $d$ fixed directions. We present an $8d/3$-approximation algorithm for this problem running in time $O((nd)^{O(d4^d)})$. The previous-best polynomial-time approximation (for constant $d$) was a classical $n^\varepsilon$ approximation by Fox and Pach [SODA'11] that has recently been improved to a $OPT^{\varepsilon}$-approximation algorithm by Cslovjecsek, Pilipczuk and W\k{e}grzycki [SODA '24], which also extends to an arbitrary set of convex polygons. Our result builds on, and generalizes the recent constant factor approximation algorithms for the maximum independent set of axis-parallel rectangles problem (which is a special case of our problem with $d=2$) by Mitchell [FOCS'21] and G\'{a}lvez, Khan, Mari, M\"{o}mke, Reddy, and Wiese [SODA'22].

翻译：在凸多边形最大独立集问题中，给定平面上 $n$ 个凸多边形集合，目标是选择最大基数的不重叠多边形子集。本文研究该问题的一个特例：多边形边的方向至多取 $d$ 个固定方向。我们提出了一个时间复杂度为 $O((nd)^{O(d4^d)})$ 的 $8d/3$-近似算法。此前对常数 $d$ 的最佳多项式时间近似是 Fox 和 Pach [SODA'11] 的经典 $n^\varepsilon$ 近似，最近由 Cslovjecsek、Pilipczuk 和 Węgrzycki [SODA '24] 改进为 $OPT^{\varepsilon}$-近似算法，该算法也适用于任意凸多边形集合。我们的结果建立在并推广了 Mitchell [FOCS'21] 及 Gálvez、Khan、Mari、Mömke、Reddy 和 Wiese [SODA'22] 针对轴平行矩形最大独立集问题（即 $d=2$ 的特例）的最新常数因子近似算法。