近线性时间及动态近似求解几何背包问题
Approximating the Geometric Knapsack Problem in Near-Linear Time and Dynamically
AI总结：
针对d维几何背包问题，论文提出近线性时间近似算法及多对数开销动态算法，并借助易猜测装填改进立方体与矩形情形的运行时间和近似保证。
摘要：
算法设计中的一个重要目标是确定（近似）求解一个问题的最佳运行时间。对于某些问题，在假定某些条件下界的情况下，我们已知最优运行时间。在本文中，我们研究d维几何背包问题，而我们对该问题的理解还远未达到这一水平。给定一组带权的d维几何物品，例如正方形、矩形或超立方体，以及一个为正方形或（超）立方体的背包。我们希望选择一个物品子集，使其互不重叠地放入背包内，并最大化装入物品的总利润。我们通过为任意常数维d和任意常数参数ε给出具有近线性运行时间的近似算法，在确定近似求解这些问题的最佳运行时间方面迈出了重要一步。对于（超）立方体，我们提出一种(1+ε)-近似算法，其运行时间大幅改进了已知的(1+ε)-近似算法；后者的运行时间中，n的指数以指数方式依赖于1/ε和d。此外，在不允许旋转的设置下，我们针对矩形给出一种(2+ε)-近似算法；若允许90度旋转，则给出一种(17/9+ε)-近似算法。这些设置下已知最佳的多项式时间算法分别具有17/9+ε和1.5+ε的近似比，并且其运行时间中n的指数以指数方式依赖于1/ε。我们还给出具有多对数查询和更新时间、且与上述算法具有相同近似保证的动态算法。我们结果的关键是一族新的结构化装填，我们称之为易猜测装填。它们足够灵活，能够保证有利可图的解，同时又具有足够的结构，使我们能够快速计算这些解。
英文摘要：
An important goal in algorithm design is determining the best running time for solving a problem (approximately). For some problems, we know the optimal running time, assuming certain conditional lower bounds. In this work, we study the $d$-dimensional geometric knapsack problem where we are far from this level of understanding. We are given a set of weighted d-dimensional geometric items like squares, rectangles, or hypercubes and a knapsack which is a square or a (hyper-)cube. We want to select a subset of items that fit non-overlappingly inside the knapsack, maximizing the total profit of the packed items. We make a significant step towards determining the best running time for solving these problems approximately by presenting approximation algorithms with near-linear running times for any constant dimension d and any constant parameter $ε$. For (hyper)-cubes, we present a $(1+ε)$-approximation algorithm whose running time drastically improves upon the known $(1+ε)$-approximation algorithm which has a running time where the exponent of n depends exponentially on $1/ε$ and $d$. Moreover, we present a $(2+ε)$-approximation algorithm for rectangles in the setting without rotations and a $(17/9+ε)$-approximation algorithm if we allow rotations by 90 degrees. The best known polynomial time algorithms for these settings have approximation ratios of $17/9+ε$ and $1.5+ε$, respectively, and running times in which the exponent of n depends exponentially on $1/ε$. We also give dynamic algorithms with polylogarithmic query and update times and the same approximation guarantees as the algorithms above. Key to our results is a new family of structured packings which we call easily guessable packings. They are flexible enough to guarantee profitable solutions and structured enough so that we can compute these solutions quickly.
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。
