We present a nearly linear work parallel algorithm for approximating the Held-Karp bound for the Metric TSP problem. Given an edge-weighted undirected graph $G=(V,E)$ on $m$ edges and $\epsilon>0$, it returns a $(1+\epsilon)$-approximation to the Held-Karp bound with high probability, in $\tilde{O}(m/\epsilon^4)$ work and $\tilde{O}(1/\epsilon^4)$ depth. While a nearly linear time sequential algorithm was known for almost a decade (Chekuri and Quanrud'17), it was not known how to simultaneously achieve nearly linear work alongside polylogarithmic depth. Using a reduction by Chalermsook et al.'22, we also give a parallel algorithm for computing a $(1+\epsilon)$-approximate fractional solution to the $k$-edge-connected spanning subgraph (kECSS) problem, with the same complexity. To obtain these results, we introduce a notion of core-sequences for the parallel Multiplicative Weights Update (MWU) framework (Luby-Nisan'93, Young'01). For the Metric TSP and kECSS problems, core-sequences enable us to exploit the structure of approximate minimum cuts to reduce the cost per iteration and/or the number of iterations. The acceleration technique via core-sequences is generic and of independent interest. In particular, it improves the best-known iteration complexity of MWU algorithms for packing/covering LPs from $poly(\log nnz(A))$ to polylogarithmic in the product of cardinalities of the core-sequence sets where $A$ is the constraint matrix of the LP. For certain implicitly defined LPs such as the kECSS LP, this yields an exponential improvement in depth.

翻译：本文提出一种近似计算度量TSP问题Held-Karp界的近线性工作并行算法。给定具有m条边的边加权无向图$G=(V,E)$及$\epsilon>0$，该算法以高概率返回Held-Karp界的$(1+\epsilon)$近似解，其工作复杂度为$\tilde{O}(m/\epsilon^4)$，深度为$\tilde{O}(1/\epsilon^4)$。尽管近线性时间顺序算法已存在近十年（Chekuri与Quanrud'17），但如何同时实现近线性工作与多对数深度此前尚未解决。利用Chalermsook等人'22的归约方法，我们还给出了计算k边连通支撑子图（kECSS）问题$(1+\epsilon)$近似分数解的并行算法，其复杂度与前者相同。为达成这些结果，我们为并行乘性权重更新（MWU）框架（Luby-Nisan'93, Young'01）引入了核心序列的概念。对于度量TSP与kECSS问题，核心序列使我们能够利用近似最小割的结构特性来降低每次迭代的代价和/或迭代次数。基于核心序列的加速技术具有通用性且具备独立研究价值。特别地，它将MWU算法求解打包/覆盖线性规划（LP）的最佳已知迭代复杂度从$poly(\log nnz(A))$改进为核心序列集合基数乘积的多对数级别，其中$A$为LP的约束矩阵。对于kECSS线性规划等隐式定义的LP问题，该技术实现了深度的指数级改进。