Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Constant-factor Approximation for Weighted Bond Cover

Eun Jung Kim, Euiwoong Lee, Dimitrios M. Thilikos

TL;DR

This work delivers a constant-factor approximation for Weighted $\mathcal{F}$-Vertex Deletion when $\mathcal{F}$ is the class of $\theta_c$-minor-free graphs, reframing the problem as Weighted $c$-Bond Cover. The algorithm leverages a structural theorem that either yields a large $c$-outgrowth, a small $\theta_c$-model, or a cluster collection that can be contracted to yield a dense graph; it combines a specialized replacer for outgrowths with a carefully designed weighting scheme in a primal-dual framework to obtain a uniformly polynomial-time $O_c(1)$-approximation. The approach simultaneously advances understanding of constant-factor approximability for minor-closed families and provides a template potentially adaptable to other classes with bounded treewidth or similar structural decompositions. These results open avenues for extending constant-factor guarantees beyond the previously known cases (Vertex Cover, Feedback Vertex Set, Diamond Hitting Set) and offer concrete tools for algorithmic graph modification problems in minor-closed families.

Abstract

The Weighted $\mathcal{F}$-Vertex Deletion for a class ${\cal F}$ of graphs asks, weighted graph $G$, for a minimum weight vertex set $S$ such that $G-S\in{\cal F}.$ The case when ${\cal F}$ is minor-closed and excludes some graph as a minor has received particular attention but a constant-factor approximation remained elusive for Weighted $\mathcal{F}$-Vertex Deletion. Only three cases of minor-closed ${\cal F}$ are known to admit constant-factor approximations, namely Vertex Cover, Feedback Vertex Set and Diamond Hitting Set. We study the problem for the class ${\cal F}$ of $θ_c$-minor-free graphs, under the equivalent setting of the Weighted $c$-Bond Cover problem, and present a constant-factor approximation algorithm using the primal-dual method. For this, we leverage a structure theorem implicit in [Joret, Paul, Sau, Saurabh, and Thomassé, SIDMA'14] which states the following: any graph $G$ containing a $θ_c$-minor-model either contains a large two-terminal protrusion, or contains a constant-size $θ_c$-minor-model, or a collection of pairwise disjoint constant-sized connected sets that can be contracted simultaneously to yield a dense graph. In the first case, we tame the graph by replacing the protrusion with a special-purpose weighted gadget. For the second and third case, we provide a weighting scheme which guarantees a local approximation ratio. Besides making an important step in the quest of (dis)proving a constant-factor approximation for Weighted $\mathcal{F}$-Vertex Deletion, our result may be useful as a template for algorithms for other minor-closed families.

A Constant-factor Approximation for Weighted Bond Cover

TL;DR

This work delivers a constant-factor approximation for Weighted -Vertex Deletion when is the class of -minor-free graphs, reframing the problem as Weighted -Bond Cover. The algorithm leverages a structural theorem that either yields a large -outgrowth, a small -model, or a cluster collection that can be contracted to yield a dense graph; it combines a specialized replacer for outgrowths with a carefully designed weighting scheme in a primal-dual framework to obtain a uniformly polynomial-time -approximation. The approach simultaneously advances understanding of constant-factor approximability for minor-closed families and provides a template potentially adaptable to other classes with bounded treewidth or similar structural decompositions. These results open avenues for extending constant-factor guarantees beyond the previously known cases (Vertex Cover, Feedback Vertex Set, Diamond Hitting Set) and offer concrete tools for algorithmic graph modification problems in minor-closed families.

Abstract

The Weighted -Vertex Deletion for a class of graphs asks, weighted graph , for a minimum weight vertex set such that The case when is minor-closed and excludes some graph as a minor has received particular attention but a constant-factor approximation remained elusive for Weighted -Vertex Deletion. Only three cases of minor-closed are known to admit constant-factor approximations, namely Vertex Cover, Feedback Vertex Set and Diamond Hitting Set. We study the problem for the class of -minor-free graphs, under the equivalent setting of the Weighted -Bond Cover problem, and present a constant-factor approximation algorithm using the primal-dual method. For this, we leverage a structure theorem implicit in [Joret, Paul, Sau, Saurabh, and Thomassé, SIDMA'14] which states the following: any graph containing a -minor-model either contains a large two-terminal protrusion, or contains a constant-size -minor-model, or a collection of pairwise disjoint constant-sized connected sets that can be contracted simultaneously to yield a dense graph. In the first case, we tame the graph by replacing the protrusion with a special-purpose weighted gadget. For the second and third case, we provide a weighting scheme which guarantees a local approximation ratio. Besides making an important step in the quest of (dis)proving a constant-factor approximation for Weighted -Vertex Deletion, our result may be useful as a template for algorithms for other minor-closed families.

Paper Structure

This paper contains 14 sections, 12 theorems, 15 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Basic definitions and preliminary results
  3. Bonds and pumpkins.
  4. Treewidth.
  5. Covering bonds.
  6. Cluster collections.
  7. Detecting and covering $c$-outgrowths.
  8. A structural theorem
  9. The weighting scheme
  10. Replacing outgrowths
  11. The primal-dual approach
  12. Discussion
  13. The case of big minimum edge-degree
  14. Necessity of a replacer

Key Result

Theorem 1

There is a constant-factor approximation algorithm for Weighted $c$-Bond Cover which runs in timeWe use notation ${\cal O}_{c}(f)$ in order to denote $g(c)\cdot {\cal O}(f),$ for some computable function $g:\Bbb{N}\to\Bbb{N}.$$\mathcal{O}_c(n^{{\cal O}(1)}),$ for every positive $c.$

Figures (2)

  • Figure 1: A visualisation of the set $W,$ the ${\cal P}$-clusters, and the ${\cal C}$-clusters in the proof of \ref{['label_indistinctement']}.
  • Figure 2: The construction of the replacement $c$-outgrowth ${\bf K}'=(K',u,v).$

Theorems & Definitions (28)

  • Theorem 1
  • Theorem 2
  • proof
  • Proposition 1
  • Proposition 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • ...and 18 more