Oriented Convex Containers of Polygons -- II

凸多边形平移装填的改进近似算法

Improved Approximations for Translational Packing of Convex Polygons

本文研究仅允许平移的二维凸多边形装填，提出基于物体变换与货架装填的多项式时间近似算法，改进面积、周长、带装填和箱装填等问题的既有保证，并推进任意多边形常数近似这一开放问题。

背包多边形装填问题的Shadoks求解方法

Shadoks Approach to Knapsack Polygonal Packing

针对2024年CG:SHOP挑战赛的背包多边形装填问题，提出结合整数线性规划、几何贪心启发式生成初始解，再经局部搜索优化的两阶段方法，最终获得赛事第一名。

Simulated annealing for weighted polygon packing

In this paper we present a new algorithm for a layout optimization problem: this concerns the placement of weighted polygons inside a circular container, the two objectives being to minimize imbalance of mass and to minimize the radius of the container. This problem carries real practical significance in industrial applications (such as the design of satellites), as well as being of significant theoretical interest. Previous work has dealt with circular or rectangular objects, but here we deal with the more realistic case where objects may be represented as polygons and the polygons are allowe…

Largest triangles in a polygon

We study the problem of finding maximum-area triangles that can be inscribed in a polygon in the plane. We consider eight versions of the problem: we use either convex polygons or simple polygons as the container; we require the triangles to have either one corner with a fixed angle or all three corners with fixed angles; we either allow reorienting the triangle or require its orientation to be fixed. We present exact algorithms for all versions of the problem. In the case with reorientations for convex polygons with $n$ vertices, we also present $(1-\varepsilon)$-approximation algorithms.

Polygon Queries for Convex Hulls of Points

We study the following range searching problem: Preprocess a set $P$ of $n$ points in the plane with respect to a set $\mathcal{O}$ of $k$ orientations % , for a constant, in the plane so that given an $\mathcal{O}$-oriented convex polygon $Q$, the convex hull of $P\cap Q$ can be computed efficiently, where an $\mathcal{O}$-oriented polygon is a polygon whose edges have orientations in $\mathcal{O}$. We present a data structure with $O(nk^3\log^2n)$ space and $O(nk^3\log^2n)$ construction time, and an $O(h+s\log^2 n)$-time query algorithm for any query $\mathcal{O}$-oriented convex $s$-gon $Q$…

Minimum-link $C$-Oriented Paths Visiting a Sequence of Regions in the Plane

Let $E=\{e_1,\ldots,e_n\}$ be a set of $C$-oriented disjoint segments in the plane, where $C$ is a given finite set of orientations that spans the plane, and let $s$ and $t$ be two points. %(We also require that for each orientation in $C$, its opposite orientation is also in $C$.) We seek a minimum-link $C$-oriented tour of $E$, that is, a polygonal path $π$ from $s$ to $t$ that visits the segments of $E$ in order, such that, the orientations of its edges are in $C$ and their number is minimum. We present an algorithm for computing such a tour in $O(|C|^2 \cdot n^2)$ time. This problem alread…