论文标题
最佳离散化是固定参数
Optimal Discretization is Fixed-parameter Tractable
论文作者
论文摘要
给定两个脱节设置$ W_1 $和$ W_2 $在飞机上的积分，最佳离散化问题要求将$ W_1 $与$ W_2 $的水平和垂直线的最小尺寸分开，也就是说，在该线路中的每个区域中，飞机都只有$ W_1 $的点，或者只有$ W_1 $的点，或者只有$ W_2 $ w_1 $ w_1 $ w_1 $ w_1 $ w_1 $ w_1 $ w_2 $ is is $ w_2 $ is空。同等地，可以将最佳离散化作为离散连续变量的一项任务：我们希望将$ x $ - 坐标的范围离散为$ y $ $ - coordinate的范围，以尽可能少的几个段为单位，并确保从$ w_1 \ times w_2 $ thime w_2 $ this Prof Propot of Choce of Prol Prop. w_1 \ timess w_2 $ w_2 $ w_2 $ w_2 $ a的范围。 我们为问题提供了固定参数算法，该算法由解决方案中的线数参数化。我们的算法在及时工作$ 2^{o（k^2 \ log k）} n^{o（1）} $，其中$ k $是要查找的行数，而$ n $是输入中的点数。 我们的结果回答了一个邦纳特，吉安诺波斯和兰皮斯[IPEC 2017]和弗罗斯（PhD论文，2018年）的正面问题，并且与两个紧密相关的概括的已知棘手性形成鲜明对比：矩形刺伤问题和所选行不需要的概括是轴线。
Given two disjoint sets $W_1$ and $W_2$ of points in the plane, the Optimal Discretization problem asks for the minimum size of a family of horizontal and vertical lines that separate $W_1$ from $W_2$, that is, in every region into which the lines partition the plane there are either only points of $W_1$, or only points of $W_2$, or the region is empty. Equivalently, Optimal Discretization can be phrased as a task of discretizing continuous variables: we would like to discretize the range of $x$-coordinates and the range of $y$-coordinates into as few segments as possible, maintaining that no pair of points from $W_1 \times W_2$ are projected onto the same pair of segments under this discretization. We provide a fixed-parameter algorithm for the problem, parameterized by the number of lines in the solution. Our algorithm works in time $2^{O(k^2 \log k)} n^{O(1)}$, where $k$ is the bound on the number of lines to find and $n$ is the number of points in the input. Our result answers in positive a question of Bonnet, Giannopolous, and Lampis [IPEC 2017] and of Froese (PhD thesis, 2018) and is in contrast with the known intractability of two closely related generalizations: the Rectangle Stabbing problem and the generalization in which the selected lines are not required to be axis-parallel.
