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An Algorithmic Weakening of the Erdős-Hajnal Conjecture

Authors Édouard Bonnet, Stéphan Thomassé, Xuan Thang Tran, Rémi Watrigant



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Édouard Bonnet
  • Univ Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR5668, France
Stéphan Thomassé
  • Univ Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR5668, France
Xuan Thang Tran
  • Univ Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR5668, France
Rémi Watrigant
  • Univ Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR5668, France

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Édouard Bonnet, Stéphan Thomassé, Xuan Thang Tran, and Rémi Watrigant. An Algorithmic Weakening of the Erdős-Hajnal Conjecture. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 23:1-23:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.ESA.2020.23

Abstract

We study the approximability of the Maximum Independent Set (MIS) problem in H-free graphs (that is, graphs which do not admit H as an induced subgraph). As one motivation we investigate the following conjecture: for every fixed graph H, there exists a constant δ > 0 such that MIS can be n^{1-δ}-approximated in H-free graphs, where n denotes the number of vertices of the input graph. We first prove that a constructive version of the celebrated Erdős-Hajnal conjecture implies ours. We then prove that the set of graphs H satisfying our conjecture is closed under the so-called graph substitution. This, together with the known polynomial-time algorithms for MIS in H-free graphs (e.g. P₆-free and fork-free graphs), implies that our conjecture holds for many graphs H for which the Erdős-Hajnal conjecture is still open. We then focus on improving the constant δ for some graph classes: we prove that the classical Local Search algorithm provides an OPT^{1-1/t}-approximation in K_{t, t}-free graphs (hence a √{OPT}-approximation in C₄-free graphs), and, while there is a simple √n-approximation in triangle-free graphs, it cannot be improved to n^{1/4-ε} for any ε > 0 unless NP ⊆ BPP. More generally, we show that there is a constant c such that MIS in graphs of girth γ cannot be n^{c/(γ)}-approximated. Up to a constant factor in the exponent, this matches the ratio of a known approximation algorithm by Monien and Speckenmeyer, and by Murphy. To the best of our knowledge, this is the first strong (i.e., Ω(n^δ) for some δ > 0) inapproximability result for Maximum Independent Set in a proper hereditary class.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Approximation algorithms analysis
Keywords
  • Approximation
  • Maximum Independent Set
  • H-free Graphs
  • Erdős-Hajnal conjecture

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References

  1. Miklós Ajtai, János Komlós, and Endre Szemerédi. A note on ramsey numbers. J. Comb. Theory, Ser. A, 29(3):354-360, 1980. URL: https://doi.org/10.1016/0097-3165(80)90030-8.
  2. S. Alekseev. A note on stable sets and colorings of graphs. Commentationes Mathematicae Universitatis Carolinae, 15, issue 2:307-309, 1974. Google Scholar
  3. V. E. Alekseev. The effect of local constraints on the complexity of determination of the graph independence number. Combinatorial-algebraic methods in applied mathematics, pages 3-13, 1982. Google Scholar
  4. Vladimir E. Alekseev. Polynomial algorithm for finding the largest independent sets in graphs without forks. Discrete Applied Mathematics, 135(1-3):3-16, 2004. URL: https://doi.org/10.1016/S0166-218X(02)00290-1.
  5. Paola Alimonti and Viggo Kann. Some apx-completeness results for cubic graphs. Theoretical Computer Science, 237(1):123-134, 2000. URL: https://doi.org/10.1016/S0304-3975(98)00158-3.
  6. Brenda S. Baker. Approximation algorithms for np-complete problems on planar graphs. J. ACM, 41(1):153-180, January 1994. URL: https://doi.org/10.1145/174644.174650.
  7. Béla Bollobás. Random Graphs, Second Edition, volume 73 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, 2011. URL: https://doi.org/10.1017/CBO9780511814068.
  8. John Adrian Bondy and Uppaluri S. R. Murty. Graph Theory. Graduate Texts in Mathematics. Springer, 2008. URL: https://doi.org/10.1007/978-1-84628-970-5.
  9. Andreas Brandstädt and Raffaele Mosca. Maximum weight independent set for lclaw-free graphs in polynomial time. Discrete Applied Mathematics, 237:57-64, 2018. URL: https://doi.org/10.1016/j.dam.2017.11.029.
  10. Timothy M. Chan. Polynomial-time approximation schemes for packing and piercing fat objects. J. Algorithms, 46(2):178-189, 2003. URL: https://doi.org/10.1016/S0196-6774(02)00294-8.
  11. Timothy M. Chan and Sariel Har-Peled. Approximation algorithms for maximum independent set of pseudo-disks. Discrete & Computational Geometry, 48(2):373-392, 2012. URL: https://doi.org/10.1007/s00454-012-9417-5.
  12. Maria Chudnovsky. The erdös-hajnal conjecture - A survey. Journal of Graph Theory, 75(2):178-190, 2014. URL: https://doi.org/10.1002/jgt.21730.
  13. Maria Chudnovsky, Marcin Pilipczuk, Michal Pilipczuk, and Stéphan Thomassé. Quasi-polynomial time approximation schemes for the maximum weight independent set problem in H-free graphs. In Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms, SODA 2020, Salt Lake City, UT, USA, January 5-8, 2020, pages 2260-2278, 2020. URL: https://doi.org/10.1137/1.9781611975994.139.
  14. E. D. Demaine, M. T. Hajiaghayi, and K. Kawarabayashi. Algorithmic graph minor theory: Decomposition, approximation, and coloring. In 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05), pages 637-646, October 2005. URL: https://doi.org/10.1109/SFCS.2005.14.
  15. Pavel Dvořák, Andreas Emil Feldmann, Ashutosh Rai, and Paweł Rzążewski. Parameterized inapproximability of independent sets in h-free graphs. In 46th International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2020), 2020. Google Scholar
  16. P. Erdős and A. Hajnal. Ramsey-type theorems. Discrete Applied Mathematics, 25(1):37-52, 1989. URL: https://doi.org/10.1016/0166-218X(89)90045-0.
  17. Paul Erdös. Graph theory and probability. ii. Canadian Journal of Mathematics, 13:346-352, 1961. Google Scholar
  18. Paul Erdős, Stephen Suen, and Peter Winkler. On the size of a random maximal graph. Random Struct. Algorithms, 6(2/3):309-318, 1995. URL: https://doi.org/10.1002/rsa.3240060217.
  19. P. Erdős. Some remarks on the theory of graphs. Bulletin of the American Mathematical Society, 53(4):292-294, 1947. URL: https://doi.org/10.1090/S0002-9904-1947-08785-1.
  20. Uriel Feige, Shafi Goldwasser, László Lovász, Shmuel Safra, and Mario Szegedy. Approximating Clique is Almost NP-Complete (Preliminary Version). In 32nd Annual Symposium on Foundations of Computer Science, San Juan, Puerto Rico, 1-4 October 1991, pages 2-12. IEEE Computer Society, 1991. URL: https://doi.org/10.1109/SFCS.1991.185341.
  21. Michael R. Garey and David S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, 1979. Google Scholar
  22. Martin Grötschel, László Lovász, and Alexander Schrijver. Geometric Algorithms and Combinatorial Optimization, volume 2 of Algorithms and Combinatorics. Springer, 1988. URL: https://doi.org/10.1007/978-3-642-97881-4.
  23. Andrzej Grzesik, Tereza Klimosova, Marcin Pilipczuk, and Michal Pilipczuk. Polynomial-time algorithm for maximum weight independent set on dollarp_6dollar-free graphs. CoRR, abs/1707.05491, 2017. URL: http://arxiv.org/abs/1707.05491.
  24. Andrzej Grzesik, Tereza Klimosova, Marcin Pilipczuk, and Michal Pilipczuk. Polynomial-time algorithm for maximum weight independent set on P₆-free graphs. In Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019, San Diego, California, USA, January 6-9, 2019, pages 1257-1271, 2019. URL: https://doi.org/10.1137/1.9781611975482.77.
  25. Magnús M. Halldórsson. Approximating discrete collections via local improvements. In Proceedings of the Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, 22-24 January 1995. San Francisco, California, USA., pages 160-169, 1995. URL: http://dl.acm.org/citation.cfm?id=313651.313687.
  26. Edin Husić, Stéphan Thomassé, and Nicolas Trotignon. The independent set problem is FPT for even-hole-free graphs. to appear in the proceedings of IPEC 2019, 2019. Google Scholar
  27. Johan Håstad. Clique is hard to approximate within n^1-epsilon. In Acta Mathematica, pages 627-636, 1996. Google Scholar
  28. Jeong Han Kim. The ramsey number r (3, t) has order of magnitude t2/log t. Random Structures & Algorithms, 7(3):173-207, 1995. Google Scholar
  29. Michael Krivelevich. Bounding ramsey numbers through large deviation inequalities. Random Struct. Algorithms, 7(2):145-156, 1995. URL: https://doi.org/10.1002/rsa.3240070204.
  30. Vadim V. Lozin and Martin Milanic. A polynomial algorithm to find an independent set of maximum weight in a fork-free graph. J. Discrete Algorithms, 6(4):595-604, 2008. URL: https://doi.org/10.1016/j.jda.2008.04.001.
  31. George J. Minty. On maximal independent sets of vertices in claw-free graphs. J. Comb. Theory, Ser. B, 28(3):284-304, 1980. URL: https://doi.org/10.1016/0095-8956(80)90074-X.
  32. Burkhard Monien and Ewald Speckenmeyer. Ramsey numbers and an approximation algorithm for the vertex cover problem. Acta Inf., 22(1):115-123, 1985. URL: https://doi.org/10.1007/BF00290149.
  33. Owen J Murphy. Computing independent sets in graphs with large girth. Discrete Applied Mathematics, 35(2):167-170, 1992. Google Scholar
  34. Nabil H. Mustafa and Saurabh Ray. Improved results on geometric hitting set problems. Discrete & Computational Geometry, 44(4):883-895, 2010. URL: https://doi.org/10.1007/s00454-010-9285-9.
  35. David Zuckerman. Linear degree extractors and the inapproximability of max clique and chromatic number. Theory of Computing, 3(1):103-128, 2007. URL: https://doi.org/10.4086/toc.2007.v003a006.
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