In this paper, we present improved approximation algorithms for the (unsplittable) Capacitated Vehicle Routing Problem (CVRP) in general metrics. In CVRP, introduced by Dantzig and Ramser (1959), we are given a set of points (clients) $V$ together with a depot $r$ in a metric space, with each $v\in V$ having a demand $d_v>0$, and a vehicle of bounded capacity $Q$. The goal is to find a minimum cost collection of tours for the vehicle, each starting and ending at the depot, such that each client is visited at least once and the total demands of the clients in each tour is at most $Q$. In the unsplittable variant we study, the demand of a node must be served entirely by one tour. We present two approximation algorithms for unsplittable CVRP: a combinatorial $(α+1.75)$-approximation, where $α$ is the approximation factor for the Traveling Salesman Problem, and an approximation algorithm based on LP rounding with approximation guarantee $α+\ln(2) + δ\approx 3.194 + δ$ in $n^{O(1/δ)}$ time. Both approximations can further be improved by a small amount when combined with recent work by Blauth, Traub, and Vygen (2021), who obtained an $(α+ 2\cdot (1 -ε))$-approximation for unsplittable CVRP for some constant $ε$ depending on $α$ ($ε> 1/3000$ for $α= 1.5$).
Approximations for Generalized Unsplittable Flow on Paths with Application to Power Systems Optimization
不可分割路径流（Unsplittable Flow on a Path，UFP）问题作为一个具有挑战性的组合优化问题，具有重要的实际意义，已经引起了人们的广泛关注。由其在电力工程中的关键应用转向，目前的工作制定了一个新的推广的UFP，其中的需求和能力在输入实例是单调阶跃函数的边缘集。作为解决这一泛化的第一步，我们借鉴和扩展的想法，从以前的研究设计一个准多项式时间近似计划（QPTAS）的前提下，要求和能力在于准多项式范围。其次，保留相同的假设，一个有效的对数近似引入的单源变量的问题。最后，我们四舍五入的贡献，通过设计一个（种）黑箱减少，在一些温和的条件下，允许翻译LP为基础的近似算法的研究问题到他们的同行交流最优潮流（AC OPF）问题-在电力系统的运行和控制的基本工作流程。
An $O(1)$-Approximation Algorithm for Dynamic Weighted Vertex Cover with Soft Capacity
本研究考虑动态环境中的（软）容量限制顶点覆盖问题。该问题推广了近年来得到广泛研究的顶点覆盖问题的动态模型。给定一个动态变化的顶点加权图$G=(V,E)$，它允许边插入和边删除，目标是设计一个数据结构，保持一个近似的最小顶点覆盖，同时满足每个顶点的容量约束。也就是说，当拾取覆盖中的顶点$v$的副本时，由副本覆盖的$v$的关联边的数量达到给定容量的$v$。我们扩展Bhattacharya等人。的工作[SODA'15和ICALP'15]，以获得一个确定性的原始-对偶算法，用于保持一个常数因子近似最小容量顶点覆盖与$O(\log n / ε)$摊销更新时间，其中$n$是在图中的顶点数。该算法可以扩展到（1）一个更一般的模型，其中每个边与一个非均匀和不可分裂的需求，（2）更一般的容量限制集覆盖问题。
