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Improved approximation ratio for covering pliable set families

Zeev Nutov

TL;DR

This work studies the Set Family Edge Cover problem under $\gamma$-pliable set families, extending beyond uncrossable families. It builds on the Goemans–Williamson primal–dual framework with a reverse-delete phase, and refines the analysis to prove a $10$-approximation for $\gamma$-pliable ${\cal F}$, improving the prior $16$-approximation. The core contribution is a tight bound derived from ${\cal F}^I$-cores, a laminar witness family, and a hollow-chain decomposition guided by Property $(\gamma)$, which yields the $10$-factor. The results also lead to better Approximation ratios for capacitated network design variants such as Near Min-Cuts Cover and Cap-$k$-ECSS, broadening the applicability of pliable-family techniques.

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.

Improved approximation ratio for covering pliable set families

TL;DR

This work studies the Set Family Edge Cover problem under -pliable set families, extending beyond uncrossable families. It builds on the Goemans–Williamson primal–dual framework with a reverse-delete phase, and refines the analysis to prove a -approximation for -pliable , improving the prior -approximation. The core contribution is a tight bound derived from -cores, a laminar witness family, and a hollow-chain decomposition guided by Property , which yields the -factor. The results also lead to better Approximation ratios for capacitated network design variants such as Near Min-Cuts Cover and Cap--ECSS, broadening the applicability of pliable-family techniques.

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 , 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 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 . This also improves approximation ratios for several variants of the Capacitated -Edge Connected Spanning Subgraph problem.
Paper Structure (3 sections, 10 theorems, 5 equations, 4 figures, 1 table)

This paper contains 3 sections, 10 theorems, 5 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['t:main']}
  3. Proof of Lemma \ref{['l:main']}

Key Result

Theorem 2

The Set Family Edge Cover problem with a $\gamma$-pliable set family ${\cal F}$ admits approximation ratio $10$.

Figures (4)

  • Figure 1: Illustration to the proof of Lemma \ref{['l:witness']}.
  • Figure 2: Illustration to Lemma \ref{['l:main']}. Black nodes are in $U$, white nodes are not in $U$, while gray nodes may or may not be in $U$.
  • Figure 3: Illustration of a shortcut of a hollow chain with $\ell=2$. Black nodes are in $U$, white nodes are not in $U$, while gray nodes may or may not be in $U$.
  • Figure 4: Illustration to the proof of Lemma \ref{['l:a']}. (i) $a_i \in C$ and $b_i \notin C$. (ii) $b_{i-1} \in C$ and one of $b_i,a_{i+1}$ not in $C$.

Theorems & Definitions (12)

  • Definition 1
  • Theorem 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6: BCGI
  • Definition 7
  • Lemma 8
  • Lemma 9
  • Lemma 10
  • ...and 2 more