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A Constant-Factor Approximation for Weighted Bond Cover

Authors Eun Jung Kim, Euiwoong Lee, Dimitrios M. Thilikos

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ACM Subject Classification
  • Mathematics of computing → Approximation algorithms
  • Mathematics of computing → Combinatorial algorithms
  • Theory of computation → Approximation algorithms analysis
  • Theory of computation → Graph algorithms analysis
Keywords
  • Constant-factor approximation algorithms
  • Primal-dual method
  • Bonds in graphs
  • Graph minors
  • Graph modification problems

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Abstract

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.

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Eun Jung Kim, Euiwoong Lee, and Dimitrios M. Thilikos. A Constant-Factor Approximation for Weighted Bond Cover. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 7:1-7:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2021.7

Author Details

Eun Jung Kim
  • Université Paris-Dauphine, PSL University, CNRS, LAMSADE, 75016, Paris, France
Euiwoong Lee
  • University of Michigan, Ann Arbor, MI, USA
Dimitrios M. Thilikos
  • LIRMM, Univ. Montpellier, CNRS, Montpellier, France

Funding

  • Kim, Eun Jung: Supported by the ANR projects ASSK (ANR-18-CE40-0025-01) and ESIGMA (ANR-17-CE23-0010) from French National Research Agency.
  • Thilikos, Dimitrios M.: Supported by the ANR projects DEMOGRAPH (ANR-16-CE40-0028), ESIGMA (ANR-17-CE23-0010), and the French-German Collaboration ANR/DFG Project UTMA (ANR-20-CE92-0027).

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