An Algorithmic Weakening of the Erdős-Hajnal Conjecture

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

Thumbnail PDF


  • Filesize: 0.63 MB
  • 18 pages

Document Identifiers

Author Details

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

Cite AsGet BibTex

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)


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
  • Approximation
  • Maximum Independent Set
  • H-free Graphs
  • Erdős-Hajnal conjecture


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Miklós Ajtai, János Komlós, and Endre Szemerédi. A note on ramsey numbers. J. Comb. Theory, Ser. A, 29(3):354-360, 1980. URL:
  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:
  5. Paola Alimonti and Viggo Kann. Some apx-completeness results for cubic graphs. Theoretical Computer Science, 237(1):123-134, 2000. URL:
  6. Brenda S. Baker. Approximation algorithms for np-complete problems on planar graphs. J. ACM, 41(1):153-180, January 1994. URL:
  7. Béla Bollobás. Random Graphs, Second Edition, volume 73 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, 2011. URL:
  8. John Adrian Bondy and Uppaluri S. R. Murty. Graph Theory. Graduate Texts in Mathematics. Springer, 2008. URL:
  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:
  10. Timothy M. Chan. Polynomial-time approximation schemes for packing and piercing fat objects. J. Algorithms, 46(2):178-189, 2003. URL:
  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:
  12. Maria Chudnovsky. The erdös-hajnal conjecture - A survey. Journal of Graph Theory, 75(2):178-190, 2014. URL:
  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:
  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:
  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:
  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:
  19. P. Erdős. Some remarks on the theory of graphs. Bulletin of the American Mathematical Society, 53(4):292-294, 1947. URL:
  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:
  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:
  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:
  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:
  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:
  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:
  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:
  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:
  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:
  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:
  35. David Zuckerman. Linear degree extractors and the inapproximability of max clique and chromatic number. Theory of Computing, 3(1):103-128, 2007. URL: