Corner Occupying Theorem for the Two-dimensional Integral Rectangle Packing Problem

矩形网格图生成树的平衡定理

Spanning-Tree Maximization Forces Isoperimetry in the Square Lattice

通过拉普拉斯乘积公式和双曲坐标，证明在固定顶点数的矩形网格图中，边长越平衡生成树数量越多，其中n×n正方形网格生成树最多。

An Existential Proof of the Conjecture on Packing Anchored Rectangles

Let $P_{n}$ be a set of $n$ points, including the origin, in the unit square $U = [0,1]^2$. We consider the problem of constructing $n$ axis-parallel and mutually disjoint rectangles inside $U$ such that the bottom-left corner of each rectangle coincides with a point in $P_{n}$ and the total area covered by the rectangles is maximized \cite{ibmpuzzle}, \cite{Winkler2007}, \cite{Winkler2010a}, \cite{Winkler2010b}. The longstanding conjecture has been that at least half of $U$ can be covered when such rectangles are properly placed. In this paper, we give an existential proof of the conjecture.

Bottom-Left Placement Theorem for Rectangle Packing

This paper proves a bottom-left placement theorem for the rectangle packing problem, stating that if it is possible to orthogonally place n arbitrarily given rectangles into a rectangular container without overlapping, then we can achieve a feasible packing by successively placing a rectangle onto a bottom-left corner in the container. This theorem shows that even for the real-parameter rectangle packing problem, we can solve it after finite times of bottom-left placement actions. Based on this theorem, we might develop efficient heuristic algorithms for solving the rectangle packing problem.