通过流网络的带桩的Tromino Tilings
Tromino Tilings with Pegs via Flow Networks
摘要：
一个tromino平铺问题是一个包装难题，其中我们给出了一个连接的格子正方形区域，我们想决定是否存在使用L形状的tromino平铺该区域。在这项工作中，我们研究了一个轻微的变化的tromino平铺问题，该地区的一些位置有钉，每个tromino带有一个洞，只能放在顶部的钉。我们提出了一个表征这个瓷砖问题与钉使用流网络，并表明（i）存在一个线性时间的最大流问题的简约减少，和（ii）计数的数量，这样的瓷砖可以在线性时间。这两个结果的证明都包含算法，然后可以用来决定在$O(n)$时间内用钉子平铺一个区域。
英文摘要：
A tromino tiling problem is a packing puzzle where we are given a region of connected lattice squares and we want to decide whether there exists a tiling of the region using trominoes with the shape of an L. In this work we study a slight variation of the tromino tiling problem where some positions of the region have pegs and each tromino comes with a hole that can only be placed on top of the pegs. We present a characterization of this tiling problem with pegs using flow networks and show that (i) there exists a linear-time parsimonious reduction to the maximum-flow problem, and (ii) counting the number of such tilings can be done in linear-time. The proofs of both results contain algorithms that can then be used to decide the tiling of a region with pegs in $O(n)$ time.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
