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A $5$-Approximation Analysis for the Cover Small Cuts Problem

Miles Simmons, Ishan Bansal, Joe Cheriyan

TL;DR

The paper studies the Cover Small Cuts problem on capacitated graphs and analyzes the WGMV primal-dual algorithm for covering pliable families of sets. By introducing symmetry and structural submodularity, it proves a strengthened property $(\gamma^{\star})$ and shows the crossing density $\rho(F)$ is at most $2$, which, via Bansal's framework, yields a $5$-approximation (better than the prior $6$-approximation). Nutov's construction is cited to demonstrate that this bound is tight up to an arbitrarily small $\varepsilon$, completing the tight analysis for this algorithm in this setting. The results highlight the critical role of symmetry and structural submodularity in improving approximation guarantees for covering pliable set families, specifically in the Cover Small Cuts context.

Abstract

In the Cover Small Cuts problem, we are given a capacitated (undirected) graph $G=(V,E,u)$ and a threshold value $λ$, as well as a set of links $L$ with end-nodes in $V$ and a non-negative cost for each link $\ell\in L$; the goal is to find a minimum-cost set of links such that each non-trivial cut of capacity less than $λ$ is covered by a link. Bansal, Cheriyan, Grout, and Ibrahimpur (arXiv:2209.11209, Algorithmica 2024) showed that the WGMV primal-dual algorithm, due to Williamson, Goemans, Mihail, and Vazirani (Combinatorica, 1995), achieves approximation ratio $16$ for the Cover Small Cuts problem; their analysis uses the notion of a pliable family of sets that satisfies a combinatorial property. Later, Bansal (arXiv:2308.15714v2, IPCO 2025) and then Nutov (arXiv:2504.03910, MFCS 2025) proved that the same algorithm achieves approximation ratio $6$. We show that the same algorithm achieves approximation ratio $5$, by using a stronger notion, namely, a pliable family of sets that satisfies symmetry and structural submodularity.

A $5$-Approximation Analysis for the Cover Small Cuts Problem

TL;DR

The paper studies the Cover Small Cuts problem on capacitated graphs and analyzes the WGMV primal-dual algorithm for covering pliable families of sets. By introducing symmetry and structural submodularity, it proves a strengthened property and shows the crossing density is at most , which, via Bansal's framework, yields a -approximation (better than the prior -approximation). Nutov's construction is cited to demonstrate that this bound is tight up to an arbitrarily small , completing the tight analysis for this algorithm in this setting. The results highlight the critical role of symmetry and structural submodularity in improving approximation guarantees for covering pliable set families, specifically in the Cover Small Cuts context.

Abstract

In the Cover Small Cuts problem, we are given a capacitated (undirected) graph and a threshold value , as well as a set of links with end-nodes in and a non-negative cost for each link ; the goal is to find a minimum-cost set of links such that each non-trivial cut of capacity less than is covered by a link. Bansal, Cheriyan, Grout, and Ibrahimpur (arXiv:2209.11209, Algorithmica 2024) showed that the WGMV primal-dual algorithm, due to Williamson, Goemans, Mihail, and Vazirani (Combinatorica, 1995), achieves approximation ratio for the Cover Small Cuts problem; their analysis uses the notion of a pliable family of sets that satisfies a combinatorial property. Later, Bansal (arXiv:2308.15714v2, IPCO 2025) and then Nutov (arXiv:2504.03910, MFCS 2025) proved that the same algorithm achieves approximation ratio . We show that the same algorithm achieves approximation ratio , by using a stronger notion, namely, a pliable family of sets that satisfies symmetry and structural submodularity.
Paper Structure (3 sections, 11 theorems, 3 figures)

This paper contains 3 sections, 11 theorems, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The crossing density of a pliable family that satisfies symmetry and structural submodularity

Key Result

Lemma 1

Let $\mathcal{F}$ be a pliable family that satisfies symmetry and structural submodularity. Then the inclusion minimal sets of $\mathcal{F}$ are (pairwise) disjoint.

Figures (3)

  • Figure 1: Illustration of property$\;(\gamma^{\star})$. The shaded region indicates $D_{j} = S_0 - (S_1 \cup\dots\cup S_{j} \cup C)$.
  • Figure 2: Illustration of Lemma \ref{['lem:disjoint']}.
  • Figure 3: Illustration of the key case of Lemma \ref{['lem:5approx']}.

Theorems & Definitions (22)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Definition 1
  • Theorem 5
  • ...and 12 more