eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-09-15
7:1
7:14
10.4230/LIPIcs.APPROX/RANDOM.2021.7
article
A Constant-Factor Approximation for Weighted Bond Cover
Kim, Eun Jung
1
Lee, Euiwoong
2
Thilikos, Dimitrios M.
3
Université Paris-Dauphine, PSL University, CNRS, LAMSADE, 75016, Paris, France
University of Michigan, Ann Arbor, MI, USA
LIRMM, Univ. Montpellier, CNRS, Montpellier, France
The Weighted ℱ-Vertex Deletion for a class ℱ of graphs asks, weighted graph G, for a minimum weight vertex set S such that G-S ∈ ℱ. The case when ℱ is minor-closed and excludes some graph as a minor has received particular attention but a constant-factor approximation remained elusive for Weighted ℱ-Vertex Deletion. Only three cases of minor-closed 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 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 et al., 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 ℱ-Vertex Deletion, our result may be useful as a template for algorithms for other minor-closed families.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol207-approx-random2021/LIPIcs.APPROX-RANDOM.2021.7/LIPIcs.APPROX-RANDOM.2021.7.pdf
Constant-factor approximation algorithms
Primal-dual method
Bonds in graphs
Graph minors
Graph modification problems