Tight analysis of the primal-dual method for edge-covering pliable set families

TL;DR

The paper analyzes the primal-dual WGMV algorithm for edge covering pliable set families, extending the classical results from uncrossable families to gamma-pliable families and related sparse instances. Through a structural, combinatorial examination of the algorithm's shortcut-tree, heavy edges, and white-chain interactions, it derives a sequence of improved approximation ratios: 7 for gamma-pliable families, 6 for sparse gamma-pliable families, and a refined 6−1/(beta+1) bound for beta-crossing sparse instances, with a lambda-edge-connected enhancement for the small-cuts case. Tightness examples accompany each bound, clarifying the limits of the current analysis. These results advance the understanding of primal-dual methods in capacitated network design variants and provide near-optimal guarantees for a broad class of edge-covering problems such as Small Cuts Cover and Steiner-type forest problems.

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. Bansal, Cheriyan, Grout, and Ibrahimpur [ICALP 2023: 15:1-15:19] showed that this algorithm achieves approximation ratio $16$ for a larger class of so called $γ$-pliable set families, that have much weaker uncrossing properties. The approximation ratio $16$ was improved to $10$ by the author [WAOA 2025: 151-166]. Recently, Bansal [arXiv:2308.15714] stated approximation ratio $8$ for $γ$-pliable families and an improved approximation ratio $5$ for an important particular case of the family of cuts of size $<k$ of a graph $H$, but his proof has an error. We will improve the approximation ratio to $7$ for the former case and give a simple proof of approximation ratio $6$ for the latter case. Furthermore, if $H$ is $λ$-edge-connected then we will show a slightly better approximation ratio $6-\frac{1}{β+1}$, where $β=\left\lfloor\frac{k-1}{\lceil(λ+1)/2\rceil}\right\rfloor$. Our analysis is supplemented by examples showing that these approximation ratios are tight for the primal-dual algorithm.

Figures (6)

• Figure 1: Illustration to the shortcut of a white chain of length $\ell=2$. Here, the black nodes belong to the same core $C$, the white node $a_1$ does not belong to any core. The weight $w(e)$ of the shortcut edge $a_0b_3$ equals to $3$ plus the number of gray nodes that belong to some core.
• Figure 2: The cases in Lemma \ref{['l:cases']}. 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 bad pair $({\cal S},{\cal S}')$ with $w({\cal S})+w({\cal S}')=7$. Blue and red nodes belong to distinct cores, while all black nodes belong to the same core.
• Figure 4: Construction of a tree ${\cal T}$ of weight $7|L|-2$ and a set of $|L|+2$ cores. (a) The shortcut tree. (b) The gadgets. (c) The laminar family and the cores.
• Figure 5: Construction of a tree ${\cal T}$ of weight $6|L|-2$ and a set of $|L|+1$ cores. Any two red nodes belong to distinct cores, while all black nodes belong to the same core. (a) The shortcut tree. (b) The gadgets. (c) The laminar family and the cores.

Theorems & Definitions (23)

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