发表机构
Roma Tre University; University of Latvia（罗马第三大学; 拉脱维亚大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究改进了基于球割分解的动态规划框架，结合曲折系统分析与快速矩阵乘法，得到了平面图连通性问题的更快亚指数算法，优化了曲折线数量上界及多个平面图问题的确定性时间复杂度，包括平面图哈密顿环问题。
AI 中文摘要
本文中，我们改进了由Dorn、Penninkx、Bodlaender和Fomin（ESA 2005）设计的基于球割分解的动态规划框架，以获得更快的平面图连通性问题的亚指数算法。我们研究这些问题与曲折线（meanders，即与固定直线交于指定数量点的简单闭合平面环）之间的关系。通过将动态规划与曲折系统分析技术以及Dorn（ESA 2006）的快速矩阵乘法技术相结合，我们得到了平面图连通性问题的改进算法。我们证明了具有2n个交叉点的曲折线数量M_n为O^*(12.806^n)，这改进了Albert和Paterson（FPSAC 2004）之前的O^*(12.901^n)上界。这进而给出了几个带多项式界权重的平面图问题的确定性时间复杂度的最佳已知经典上界，即平面图旅行商问题为O(2^{5.543√n})，平面图最长环/路径为O(2^{5.796√n})，平面图连通支配集为O(2^{8.251√n})，平面图斯坦纳树为O(2^{8.037√n})。值得注意的是，这得到了平面图哈密顿环问题的最佳已知确定性复杂度O(2^{5.543√n})。
英文摘要
In this paper, we refine the dynamic programming framework based on the sphere cut decomposition designed by Dorn, Penninkx, Bodlaender, and Fomin (ESA 2005) to obtain faster subexponential algorithms for connectivity problems on planar graphs. We investigate the relationship between these problems and meanders, which are simple closed planar loops that intersect a fixed line in a given number of points. By combining dynamic programming with techniques from meandric system analysis and the use of fast matrix multiplication by Dorn (ESA 2006), we obtain improved algorithms for planar connectivity problems. We show that the number of meanders on $2n$ crossings $M_n$ is $\mathcal O^*(12.806^n)$, which improves the previous upper bound of $\mathcal O^*(12.901^n)$ by Albert and Paterson (FPSAC 2004). This then gives the best-known classical upper bounds on the deterministic time complexity of several planar graph problems with polynomially-bounded weights, namely $\mathcal O(2^{5.543\sqrt n})$ for the Planar Travelling Salesman problem, $\mathcal O(2^{5.796\sqrt n})$ for Planar Longest Cycle/Path, $\mathcal O(2^{8.251\sqrt n})$ for Planar Connected Dominating Set and $\mathcal O(2^{8.037\sqrt n})$ for Planar Steiner Tree. Notably, this leads to the best-known deterministic complexity $\mathcal O(2^{5.543\sqrt{n}})$ for the Planar Hamiltonian Cycle problem.
