Improved approximation ratio for covering pliable set families

TL;DR

This paper improves the approximation ratio for covering pliable set families from 16 to 10, refining previous algorithms and extending results to related capacitated spanning subgraph problems.

Contribution

It refines the analysis of existing algorithms to achieve a better approximation ratio for covering pliable set families, extending to related network design problems.

Findings

Approximation ratio improved from 16 to 10.

Refined analysis applies to broader set family classes.

Results impact capacitated k-edge connected spanning subgraph variants.

Abstract

A classic result of Williamson, Goemans, Mihail, and Vazirani [STOC 1993: 708-717] states that the problem of covering an uncrossable set family by a min-cost edge set admits approximation ratio $2$, by a primal-dual algorithm with a reverse delete phase. Recently, Bansal, Cheriyan, Grout, and Ibrahimpur [ICALP 2023: 15:1-15:19] showed that this algorithm achieves approximation ratio $16$ for a larger class of set families, that have much weaker uncrossing properties. In this paper we will refine their analysis and show an approximation ratio of $10$. This also improves approximation ratios for several variants of the Capacitated $k$-Edge Connected Spanning Subgraph problem.