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

TL;DR

The paper generalizes the primal-dual framework of Williamson et al. from uncrossable to pliable $f$-connectivity functions that satisfy property $(oldsymbol{iggamma})$, achieving a robust $16$-approximation for this broader class. It then leverages this main result to obtain improved guarantees for three network-design problems: AugSmallCuts (16-approximation), Cap-$k$-ECSS (16 · ⌈k/u_min⌉-approximation), and $(p,2)$-FGC (O(1)-approximation, specifically 20, with a 6-approximation for even $p$). The analysis hinges on a laminar witness structure induced by a token-distribution argument, enabled by property $(oldsymbol{iggamma})$, which substitutes for the laminar-dual facility in uncrossable settings. Overall, the work broadens the applicability and effectiveness of primal-dual methods in network design, yielding constant-factor algorithms for several practically relevant connectivity augmentation problems.

Abstract

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.

Figures (4)

• Figure 1: Illustration of property $(\gamma)$, and the sets $C, S_1, S_2$. The set $S_2 \setminus (S_1\cup C)$ is non-empty.
• Figure 2: Illustration of the witness sets $S_1 \subsetneq S_2 \subsetneq S_3 \subsetneq S_4$ and the edges $a_ib_i\; (i=1,\dots,4)$, in the proof of Lemma \ref{['lem:colorednodes']}.
• Figure 3: Top: A bad example for the primal-dual method for augmenting a pliable family. Bottom: Edges of type (A) and type (B) in a bad example for the primal-dual method for augmenting a pliable family.
• Figure 4: An instance of the $\mathrm{AugSmallCuts}$ problem where every optimal dual solution has non-laminar support.

Theorems & Definitions (39)

• Definition 1.1: WGMV95
• Definition 1.2
• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• proof
• proof
• Lemma 2.3
• proof