通过将原始-对偶方法推广至非交叉函数之外改进近似算法
Improved Approximation Algorithms by Generalizing the Primal-Dual Method Beyond Uncrossable Functions
AI总结：
本文将原始-对偶方法推广至非交叉函数之外，证明了Williamson等人的算法对更广泛的函数类达到16近似比，并给出了网络设计问题的三个应用。
摘要：
我们解决了Williamson, Goemans, Vazirani和Mihail提出的关于通过原始-对偶方法（Combinatorica 15(3):435-454, 1995）设计网络设计问题近似算法的长期悬而未决的问题。Williamson等人证明了对于连通性需求可由所谓的非交叉函数指定的连通性增强问题，其近似保证为2。他们指出：“扩展我们的算法以处理非非交叉函数仍然是一个具有挑战性的开放问题。非交叉函数的关键特征是存在一个层状的最优对偶解。这一性质刻画了非交叉函数……一个更大的开放问题是进一步探索原始-对偶方法在获取其他组合优化问题近似算法方面的能力。”我们的主要结果证明了Williamson等人的原始-对偶算法对于一类推广了非交叉函数概念的函数达到了16的近似比。存在可用我们方法处理的实例，其中没有任何最优对偶解具有层状支撑。我们展示了主要结果的三个应用。（1）用于增强图G的一族小割的16-近似算法。（2）用于Cap-k-ECSS问题的16·⌈k/umin⌉-近似算法，该问题如下：给定一个具有边成本c∈Q≥0^E和边容量u∈Z≥0^E的无向图G=(V,E)，寻找边的一个最小成本子集F⊆E，使得(V,F)中任何割的容量至少为k；我们用umin表示E中边的最小容量。（3）用于(p,2)-Flexible Graph Connectivity模型的O(1)-近似算法。
英文摘要：
We address long-standing open questions raised by Williamson, Goemans, Vazirani and Mihail pertaining to the design of approximation algorithms for problems in network design via the primal-dual method (Combinatorica 15(3):435-454, 1995). Williamson et al. prove an approximation guarantee of two for connectivity augmentation problems where the connectivity requirements can be specified by so-called uncrossable functions. They state: ``Extending our algorithm to handle non-uncrossable functions remains a challenging open problem. The key feature of uncrossable functions is that there exists an optimal dual solution which is laminar. This property characterizes uncrossable functions\dots\ A larger open issue is to explore further the power of the primal-dual approach for obtaining approximation algorithms for other combinatorial optimization problems.'' Our main result proves that the primal-dual algorithm of Williamson et al. achieves an approximation ratio of 16 for a class of functions that generalizes the notion of an uncrossable function. There exist instances that can be handled by our methods where none of the optimal dual solutions has a laminar support. We present three applications of our main result. (1) A 16-approximation algorithm for augmenting a family of small cuts of a graph $G$. (2) A $16 \cdot {\lceil k/u_{min} \rceil}$-approximation algorithm for the Cap-$k$-ECSS problem which is as follows: Given an undirected graph $G = (V,E)$ with edge costs $c \in \mathbb{Q}_{\geq 0}^E$ and edge capacities $u \in \mathbb{Z}_{\geq 0}^E$, find a minimum-cost subset of the edges $F\subseteq E$ such that the capacity of any cut in $(V,F)$ is at least $k$; we use $u_{min}$ to denote the minimum capacity of an edge in $E$. (3) An $O(1)$-approximation algorithm for the model of $(p,2)$-Flexible Graph Connectivity.
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