打破最大加权 $3$-集合打包的 $\sqrt{3}$ 障碍
Breaking the $\sqrt{3}$ Barrier for Maximum Weighted $3$-Set Packing
浏览论文内容
中文总结 AI 辅助
本文提出一种确定性多项式时间 $1.6908$ 近似算法，通过两阶段组合平方权重与原始目标，打破最大加权 $3$-集合打包的 $\sqrt{3}$ 局部性间隙障碍，显著改进此前最优界。
中文摘要 AI 辅助
我们给出了最大加权 $3$-集合打包问题的确定性多项式时间 $1.6908$ 近似算法，打破了平方权重局部搜索的 $\sqrt3$ 局部性间隙障碍。该问题的近似比从 Berman 的 $2$ [Ber00] 进展到 Neuwohner 的 $2-\frac{1}{63{,}700{,}992}+\epsilon$ [Neu21]。随后 Thiery 和 Ward 获得了 $1.786$ [TW23]，而 Thiery 进一步将界改进到 $1.761+\epsilon$，并最终通过分层交换分析达到 $\sqrt3 \approx 1.732051$ [Thi23]。Thiery 还证明了即使允许任意大小的交换，$\sqrt3$ 也是平方权重目标的局部性间隙下界。我们的算法分两个阶段执行，并组合两个目标。阶段 I 计算平方权重势的有界交换局部最优解，并通过 Thiery 的分层框架进行分析，同时通过保留非单例组件的内部树边松弛并利用 $3$-集合的关联结构来处理最终单例，从而加强终端分析。这产生了一个增强的结构不等式，在原始目标下具有残余的正爪增益。阶段 II 切换到原始目标，并通过一个辅助的加权 $9$-集合打包实例恢复足够的残余增益。一个覆盖论证将结构界通过仅用于分析的高周长提升传递。
英文摘要
We give a deterministic polynomial-time $1.6908$-approximation for Maximum Weighted $3$-Set Packing, breaking the $\sqrt3$ locality-gap barrier of squared-weight local search. The approximation ratio for this problem progressed from Berman's $2$ [Ber00] to Neuwohner's $2-\frac{1}{63{,}700{,}992}+ε$ [Neu21]. Thiery and Ward then obtained $1.786$ [TW23], while Thiery subsequently improved the bound to $1.761+ε$ and finally to $\sqrt3 \approx 1.732051$ through a layered exchange analysis [Thi23]. Thiery also proved that $\sqrt3$ is a locality-gap lower bound for the squared-weight objective even with exchanges of arbitrary size. Our algorithm performs in two phases and combines two objectives. Phase~I computes a bounded-exchange local optimum for the squared-weight potential and analyzes it through Thiery's layered framework, while strengthening the terminal analysis by preserving internal tree-edge slack for nonsingleton components and exploiting the incidence structure of $3$-sets for final singletons. This yields an augmented structural inequality with residual positive claw gain under the original objective. Phase~II switches to the original objective and recovers sufficient residual gain through an auxiliary weighted $9$-Set Packing instance. A covering argument transfers the structural bound through the high-girth lift used only in the analysis.
发表机构
- Georgia Southern University（佐治亚南方大学）
机构由 AI 辅助整理，请以论文原文为准。
