一个2.37332竞争的在线正方形装箱重力算法

中文摘要 AI 辅助

我们考虑在俄罗斯方块和重力约束下，将轴平行正方形在线装箱到单位宽度条带中的问题：一个到达的正方形必须从上方沿单调下降路径降低，直到它从下方获得支撑。Fekete、Kamphans和Schweer [Algorithmica, 2014]在该模型中给出了渐近竞争比为$34/13\approx2.6154$的算法。我们提出$\mathrm{AsymmetricSlots}$，一种基于将每个槽位划分为宽子槽位和窄子槽位的递归算法。证明使用局部收费论证：相对于其关联槽位较大的正方形用其自身面积支付它们产生的高度，而较小的正方形在两个子槽位之间平衡，并可能使用有限的有界临时信用。对于合适的划分参数$p^\star$，我们证明对于每个输入序列$\sigma$，$\mathrm{AsymmetricSlots}{p^\star}(\sigma)\le 2.37332 \operatorname{OPT} (\sigma)+O(1)$。此外，我们表明相同的框架给出了对于宽高比至多为$\kappa$的矩形，渐近竞争比为$O(\kappa)$的算法，并且匹配的$\Omega(\kappa)$下界表明对$\kappa$的依赖是渐近最优的。对于正方形算法，我们给出其渐近竞争比为$2$的下界。

英文摘要

We consider online packing of axis-parallel squares into a unit-width strip under the Tetris and gravity constraints: An incoming square must be lowered from above along a monotonic downwards path until it reaches support from below. Fekete, Kamphans, and Schweer [Algorithmica, 2014] gave an algorithm with asymptotic competitive ratio $34/13\approx2.6154$ in this model. We present $\mathrm{AsymmetricSlots}$, a recursive algorithm based on splitting each slot into a wide and narrow subslot. The proof uses a local charging argument: squares that are large relative to its associated slot pay for the height they create with their own area, while smaller squares are balanced between the two subslots and may use a bounded temporary credit. For a suitable split parameter $p^\star$, we prove $\mathrm{AsymmetricSlots}{p^\star}(σ)\le 2.37332 \operatorname{OPT} (σ)+O(1)$ for every input sequence $σ$. Additionally, we show that the same framework gives an algorithm with asymptotic competitive ratio $O(κ)$ for rectangles of aspect ratio at most $κ$, and a matching $ Ω(κ)$ lower bound shows that the dependence on $κ$ is asymptotically optimal. For the square algorithm, we give a lower bound of $2$ on its asymptotic competitive ratio.

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13 pages. 5 figures