Improved Approximation Algorithms by Generalizing the Primal-Dual Method Beyond Uncrossable Functions

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 (Williamson et al. in Combinatorica 15(3):435–454, 1995. https://doi.org/10.1007/BF01299747). Williamson et al. prove an approximation ratio of two for connectivity augmentation problems where the connectivity requirements can be specified by 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 ... 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 \(16\) $16$ 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 to problems in the area of network design. (1)  A \(16\) $16$ -approximation algorithm for augmenting a family of small cuts of a graph G. The previous best approximation ratio was \(O(\log {|V(G)|})\) $O(log|V(G)|)$ . (2)  A \(16\cdot {\lceil k/u_{min} \rceil }\) $16·\lceil k/u_{\mathit{min}}\rceil$ -approximation algorithm for the Cap-k-ECSS problem which is as follows: Given an undirected graph \(G = (V,E)\) $G=(V,E)$ with edge costs \(c \in {\mathbb {Q}}_{\ge 0}^E\) $c\in {\mathbb{Q}}_{\ge 0}^E$ and edge capacities \(u \in {\mathbb {Z}}_{\ge 0}^E\) $u\in {\mathbb{Z}}_{\ge 0}^E$ , find a minimum-cost subset of the edges \(F\subseteq E\) $F\subseteq E$ such that the capacity of any cut in (V, F) is at least k; \(u_{min}\) $u_{\mathit{min}}$ (respectively, \(u_{max}\) $u_{\mathit{max}}$ ) denotes the minimum (respectively, maximum) capacity of an edge in E, and w.l.o.g. \(u_{max} \le k\) $u_{\mathit{max}}\le k$ . The previous best approximation ratio was \(\min (O(\log {|V|}), k, 2u_{max})\) $min(O(log|V|),k,2u_{\mathit{max}})$ . (3)  A \(20\) $20$ -approximation algorithm for the model of (p, 2)-Flexible Graph Connectivity. The previous best approximation ratio was \(O(\log {|V(G)|})\) $O(log|V(G)|)$ , where G denotes the input graph.