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 (Combinatorica 15(3):435-454, 1995). 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 a 16-approximation ratio via the primal-dual method 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 have a laminar support. We present applications of our main result to three network-design problems. 1. A 16-approximation algorithm for augmenting the family of small cuts of a graph G. The previous best approximation ratio was O(log |V (G)|). 2. A 16 · ⌈k/umin⌉-approximation algorithm for the Cap-k-ECSS problem which is as follows: Given an undirected graph G = (V, E) with edge costs c ∈ QE≥0 and edge capacities u ∈ ZE≥0, find a minimum cost subset of the edges F ⊆ E such that the capacity across any cut in (V, F) is at least k; umin (respectively, umax) denote the minimum (respectively, maximum) capacity of an edge in E, and w.l.o.g. umax ≤ k. The previous best approximation ratio was min(O(log |V |), k, 2umax). 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.