针对小树宽线性规划的近线性时间算法：鲁棒中心路径的多尺度表示
A Nearly-Linear Time Algorithm for Linear Programs with Small Treewidth: A Multiscale Representation of Robust Central Path
AI总结：
针对小树宽线性规划问题，本文结合内点法、sketching及多尺度基下的解表示，提出近线性时间算法，证明了Fom等人的猜想，运行时间几乎匹配求解子问题的最快速度。
摘要：
源于结构图论，树宽已成为组合数学、整数线性规划和数值分析等多个社区中固定参数可解算法的研究焦点。已知许多 NP-难问题可在 $\widetilde{O}(n \cdot 2^{O(\mathrm{tw})})$ 时间内求解，其中 $\mathrm{tw}$ 是输入图的树宽。类似地，P 中的许多问题应能在 $\widetilde{O}(n \cdot \mathrm{tw}^{O(1)})$ 时间内求解；然而，由于缺乏合适的工具，目前仅有少数此类结果。[Fom+18] 猜想这适用于所有线性规划；在本文中，我们证明这是正确的：给定形如 $\min_{Ax=b,\ell \leq x\leq u} c^{\top} x$ 的线性规划，以及与 $A$ 相关的图 $G_A$ 的宽度为 $τ$ 的树分解，我们展示了如何在 $$\widetilde{O}(n \cdot τ^2 \log (1/\varepsilon))$$ 时间内求解它，其中 $n$ 是变量数，$\varepsilon$ 是相对精度。结合顶点容量流 [BGS21] 的最新技术，这导出了一个运行时间为 $\widetilde{O}(n^{1+o(1)} \cdot \mathrm{tw}^2 \log (1/\varepsilon))$ 的算法。除了是首创之外，我们的算法运行时间几乎匹配求解子问题 $Ax=b$ 的最快运行时间（假设不使用快速矩阵乘法）。我们通过结合内点法（IPMs）、sketching 以及在类似于小波基的多尺度基下解的新颖表示来获得这些结果。
英文摘要：
Arising from structural graph theory, treewidth has become a focus of study in fixed-parameter tractable algorithms in various communities including combinatorics, integer-linear programming, and numerical analysis. Many NP-hard problems are known to be solvable in $\widetilde{O}(n \cdot 2^{O(\mathrm{tw})})$ time, where $\mathrm{tw}$ is the treewidth of the input graph. Analogously, many problems in P should be solvable in $\widetilde{O}(n \cdot \mathrm{tw}^{O(1)})$ time; however, due to the lack of appropriate tools, only a few such results are currently known. [Fom+18] conjectured this to hold as broadly as all linear programs; in our paper, we show this is true: Given a linear program of the form $\min_{Ax=b,\ell \leq x\leq u} c^{\top} x$, and a width-$τ$ tree decomposition of a graph $G_A$ related to $A$, we show how to solve it in time $$\widetilde{O}(n \cdot τ^2 \log (1/\varepsilon)),$$ where $n$ is the number of variables and $\varepsilon$ is the relative accuracy. Combined with recent techniques in vertex-capacitated flow [BGS21], this leads to an algorithm with $\widetilde{O}(n^{1+o(1)} \cdot \mathrm{tw}^2 \log (1/\varepsilon))$ run-time. Besides being the first of its kind, our algorithm has run-time nearly matching the fastest run-time for solving the sub-problem $Ax=b$ (under the assumption that no fast matrix multiplication is used). We obtain these results by combining recent techniques in interior-point methods (IPMs), sketching, and a novel representation of the solution under a multiscale basis similar to the wavelet basis.
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。
