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Approximation Algorithms for Partially Colorable Graphs

Authors Suprovat Ghoshal, Anand Louis, Rahul Raychaudhury



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Author Details

Suprovat Ghoshal
  • Indian Institute of Science, Bangalore, India
Anand Louis
  • Indian Institute of Science, Bangalore, India
Rahul Raychaudhury
  • Indian Institute of Science, Bangalore, India

Acknowledgements

The first author thanks Pasin Manurangsi for pointing him to the Odd Cycle Transversal problem.

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Suprovat Ghoshal, Anand Louis, and Rahul Raychaudhury. Approximation Algorithms for Partially Colorable Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 28:1-28:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2019.28

Abstract

Graph coloring problems are a central topic of study in the theory of algorithms. We study the problem of partially coloring partially colorable graphs. For alpha <= 1 and k in Z^+, we say that a graph G=(V,E) is alpha-partially k-colorable, if there exists a subset S subset V of cardinality |S| >= alpha |V| such that the graph induced on S is k-colorable. Partial k-colorability is a more robust structural property of a graph than k-colorability. For graphs that arise in practice, partial k-colorability might be a better notion to use than k-colorability, since data arising in practice often contains various forms of noise. We give a polynomial time algorithm that takes as input a (1 - epsilon)-partially 3-colorable graph G and a constant gamma in [epsilon, 1/10], and colors a (1 - epsilon/gamma) fraction of the vertices using O~(n^{0.25 + O(gamma^{1/2})}) colors. We also study natural semi-random families of instances of partially 3-colorable graphs and partially 2-colorable graphs, and give stronger bi-criteria approximation guarantees for these family of instances.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Approximation algorithms
Keywords
  • Approximation Algorithms
  • Vertex Coloring
  • Semi-random Models

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