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 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 -approximation algorithm for augmenting a family of small cuts of a graph G. The previous best approximation ratio was O(log|V(G)|) . (2) A 16·⌈ k/u_min⌉ -approximation algorithm for the Cap-k-ECSS problem which is as follows: Given an undirected graph G = (V,E) with edge costs c ∈ℚ_≥ 0^E and edge capacities u ∈ℤ_≥ 0^E , find a minimum-cost subset of the edges F⊆ E such that the capacity of any cut in (V, F) is at least k; u_min (respectively, u_max ) denotes the minimum (respectively, maximum) capacity of an edge in E, and w.l.o.g. u_max≤ k . The previous best approximation ratio was min (O(log|V|), k, 2u_max) . (3) A 20 -approximation algorithm for the model of (p, 2)-Flexible Graph Connectivity. The previous best approximation ratio was O(log|V(G)|) , where G denotes the input graph.