Skip to main content
SpringerLink
Log in

Independent Set in k-Claw-Free Graphs: Conditional \(\chi \)-Boundedness and the Power of LP/SDP Relaxations

  • Conference paper
  • First Online:
Approximation and Online Algorithms (WAOA 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14297))

Included in the following conference series:

Abstract

This paper studies k-claw-free graphs, exploring the connection between an extremal combinatorics question and the power of a convex program in approximating the maximum-weight independent set in this graph class. For the extremal question, we consider the notion, that we call conditional \(\chi \)-boundedness of a graph: Given a graph G that is assumed to contain an independent set of a certain (constant) size, we are interested in upper bounding the chromatic number in terms of the clique number of G. This question, besides being interesting on its own, has algorithmic implications (which have been relatively neglected in the literature) on the performance of SDP relaxations in estimating the value of maximum-weight independent set.

For \(k=3\), Chudnovsky and Seymour (JCTB 2010) prove that any 3-claw-free graph G with an independent set of size three must satisfy \(\chi (G) \le 2 \omega (G)\). Their result implies a factor 2-estimation algorithm for the maximum weight independent set via an SDP relaxation (providing the first non-trivial result for maximum-weight independent set in such graphs via a convex relaxation). An obvious open question is whether a similar conditional \(\chi \)-boundedness phenomenon holds for any k-claw-free graph. Our main result answers this question negatively. We further present some evidence that our construction could be useful in studying more broadly the power of convex relaxations in the context of approximating maximum weight independent set in k-claw free graphs. In particular, we prove a lower bound on families of convex programs that are stronger than known convex relaxations used algorithmically in this context.

This research was partially done during the trimester on Discrete Optimization at Hausdorff Research Institute for Mathematics (HIM) in Bonn, Germany. The research has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 759557) and from Academy of Finland (grant number 310415). Kamyar Khodamoradi was supported by Deutsche Forschungsgemeinschaft (project number 399223600).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 49.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 64.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Recall that the polytope is defined as \(\textsf{QSTAB}(G)= \{x \in [0,1]^{|V(G)|}: \sum _{i \in Q} x_i \le 1 (\forall \text {clique Q})\}\). Optimizing this itself is NP-hard, but we can optimize an SDP whose solution is feasible for QSTAB.

  2. 2.

    The notation \(\widetilde{\varOmega }\) hides asymptotically smaller terms.

  3. 3.

    A clique replacement operation on graph G replaces any vertex v with a clique \(K_v\) of arbitrary size and connects each vertex in a clique to every neighbor of v. It is easy to see that k-claw-free graphs are closed under clique replacement.

  4. 4.

    The notations \(O_k, \varTheta _k, \varOmega _k\) hide multiplicative functions in k.

  5. 5.

    In fact, the hardness even holds for a special case of the problem, namely the unweighted k-set packing problem.

  6. 6.

    Consider the clique \(K_n\) on n vertices and LP assignment \(x_i=1/2\) for vertex \(i \in K_n\).

References

  1. Austrin, P., Khot, S., Safra, M.: Inapproximability of vertex cover and independent set in bounded degree graphs. Theory Comput. 7(1), 27–43 (2011)

    Article  MathSciNet  Google Scholar 

  2. Bansal, N., Gupta, A., Guruganesh, G.: On the Lovász theta function for independent sets in sparse graphs. In: Proceedings of the Forty-Seventh Annual ACM Symposium on Theory of Computing, pp. 193–200 (2015)

    Google Scholar 

  3. Berman, P.: A d/2 approximation for maximum weight independent set in \(d\)-claw free graphs. In: SWAT 2000. LNCS, vol. 1851, pp. 214–219. Springer, Heidelberg (2000). https://doi.org/10.1007/3-540-44985-X_19

    Chapter  Google Scholar 

  4. Bohman, T.: The triangle-free process. Adv. Math. 221(5), 1653–1677 (2009). https://doi.org/10.1016/j.aim.2009.02.018. https://www.sciencedirect.com/science/article/pii/S0001870809000620

  5. Bohman, T., Keevash, P.: The early evolution of the h-free process. Invent. Math. 181(2), 291–336 (2010)

    Article  MathSciNet  Google Scholar 

  6. Chalermsook, P., Chuzhoy, J.: Maximum independent set of rectangles. In: Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 892–901. SIAM (2009)

    Google Scholar 

  7. Chalermsook, P., Gadekar, A., Khodamoradi, K., Spoerhase, J.: Independent set in \(k\)-claw-free graphs: conditional \(\chi \)-boundedness and the power of LP/SDP relaxations (2023)

    Google Scholar 

  8. Chalermsook, P., Vaz, D.: A note on fractional coloring and the integrality gap of LP for maximum weight independent set. Electron. Notes Discret. Math. 55, 113–116 (2016)

    Article  Google Scholar 

  9. Chalermsook, P., Walczak, B.: Coloring and maximum weight independent set of rectangles. In: Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 860–868. SIAM (2021)

    Google Scholar 

  10. Chan, T.M., Har-Peled, S.: Approximation algorithms for maximum independent set of pseudo-disks. In: Proceedings of the Twenty-Fifth Annual Symposium on Computational Geometry, pp. 333–340 (2009)

    Google Scholar 

  11. Chan, Y.H., Lau, L.C.: On Linear and Semidefinite Programming Relaxations for Hypergraph Matching, pp. 1500–1511. https://doi.org/10.1137/1.9781611973075.122. https://epubs.siam.org/doi/abs/10.1137/1.9781611973075.122

  12. Chandra, B., Halldórsson, M.M.: Greedy local improvement and weighted set packing approximation. J. Algorithms 39(2), 223–240 (2001)

    Article  MathSciNet  Google Scholar 

  13. Chudnovsky, M., Seymour, P.: Claw-free graphs VI. colouring. J. Comb. Theory Ser. B 100(6), 560–572 (2010). https://doi.org/10.1016/j.jctb.2010.04.005. https://www.sciencedirect.com/science/article/pii/S009589561000064X

  14. Cygan, M.: Improved approximation for 3-dimensional matching via bounded pathwidth local search. In: 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, pp. 509–518. IEEE (2013)

    Google Scholar 

  15. Fleming, N., Kothari, P., Pitassi, T., et al.: Semialgebraic proofs and efficient algorithm design. Found. Trends® Theor. Comput. Sci. 14(1–2), 1–221 (2019)

    Google Scholar 

  16. Grötschel, M., Lovász, L., Schrijver, A.: Polynomial algorithms for perfect graphs. In: North-Holland Mathematics Studies, vol. 88, pp. 325–356. Elsevier (1984)

    Google Scholar 

  17. Hazan, E., Safra, S., Schwartz, O.: On the complexity of approximating k-set packing. Comput. Complex. 15(1), 20–39 (2006)

    Article  MathSciNet  Google Scholar 

  18. Hochbaum, D.S.: Efficient bounds for the stable set, vertex cover and set packing problems. Discret. Appl. Math. 6(3), 243–254 (1983)

    Article  MathSciNet  Google Scholar 

  19. Kim, J.H.: The Ramsey number \(R(3, t)\) has order of magnitude \(t^2/\log t\). Random Struct. Algorithms 7(3), 173–207 (1995)

    Article  MathSciNet  Google Scholar 

  20. Lasserre, J.B.: An explicit exact SDP relaxation for nonlinear 0-1 programs. In: Aardal, K., Gerards, B. (eds.) IPCO 2001. LNCS, vol. 2081, pp. 293–303. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-45535-3_23

    Chapter  Google Scholar 

  21. Lasserre, J.B.: Global optimization with polynomials and the problem of moments. SIAM J. Optim. 11(3), 796–817 (2001)

    Article  MathSciNet  Google Scholar 

  22. Laurent, M.: A comparison of the sherali-adams, lovász-schrijver, and lasserre relaxations for 0–1 programming. Math. Oper. Res. 28(3), 470–496 (2003)

    Article  MathSciNet  Google Scholar 

  23. Lewin-Eytan, L., Naor, J.S., Orda, A.: Routing and admission control in networks with advance reservations. In: Jansen, K., Leonardi, S., Vazirani, V. (eds.) APPROX 2002. LNCS, vol. 2462, pp. 215–228. Springer, Heidelberg (2002). https://doi.org/10.1007/3-540-45753-4_19

    Chapter  Google Scholar 

  24. Lovász, L.: A characterization of perfect graphs. J. Comb. Theory Ser. B 13(2), 95–98 (1972)

    Article  MathSciNet  Google Scholar 

  25. Neuwohner, M.: An improved approximation algorithm for the maximum weight independent set problem in d-claw free graphs. In: Bläser, M., Monmege, B. (eds.) 38th International Symposium on Theoretical Aspects of Computer Science, STACS 2021, Saarbrücken, Germany, 16–19 March 2021 (Virtual Conference). LIPIcs, vol. 187, pp. 53:1–53:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021). https://doi.org/10.4230/LIPIcs.STACS.2021.53

  26. Parrilo, P.A.: Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization. California Institute of Technology (2000)

    Google Scholar 

  27. Parrilo, P.A.: Semidefinite programming relaxations for semialgebraic problems. Math. Program. 96, 293–320 (2003)

    Article  MathSciNet  Google Scholar 

  28. Rothvoß, T.: The Lasserre hierarchy in approximation algorithms. Lecture Notes for the MAPSP, pp. 1–25 (2013)

    Google Scholar 

  29. Scott, A., Seymour, P.: A survey of \(\chi \)-boundedness. J. Graph Theory 95(3), 473–504 (2020)

    Article  MathSciNet  Google Scholar 

  30. Sherali, H.D., Adams, W.P.: A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems. SIAM J. Discret. Math. 3(3), 411–430 (1990)

    Article  MathSciNet  Google Scholar 

  31. Thiery, T., Ward, J.: An improved approximation for maximum weighted k-set packing. In: Bansal, N., Nagarajan, V. (eds.) Proceedings of the 2023 ACM-SIAM Symposium on Discrete Algorithms, SODA 2023, Florence, Italy, 22–25 January 2023, pp. 1138–1162. SIAM (2023). https://doi.org/10.1137/1.9781611977554.ch42

  32. Yang, J.: On coloring claw-free graphs (2007). https://www.mathcs.bethel.edu/yang/papers/clawfree.pdf

Download references

Author information

Authors and Affiliations

  1. Aalto University, Espoo, Finland

    Parinya Chalermsook & Ameet Gadekar

  2. University of Regina, Regina, Canada

    Kamyar Khodamoradi

  3. University of Sheffield, Sheffield, UK

    Joachim Spoerhase

Authors
  1. Parinya Chalermsook
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ameet Gadekar
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Kamyar Khodamoradi
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Joachim Spoerhase
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ameet Gadekar .

Editor information

Editors and Affiliations

  1. University of Wrocław, Wrocław, Poland

    Jarosław Byrka

  2. Technical University of Munich, Munich, Germany

    Andreas Wiese

Rights and permissions

Reprints and permissions

Copyright information

© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Chalermsook, P., Gadekar, A., Khodamoradi, K., Spoerhase, J. (2023). Independent Set in k-Claw-Free Graphs: Conditional \(\chi \)-Boundedness and the Power of LP/SDP Relaxations. In: Byrka, J., Wiese, A. (eds) Approximation and Online Algorithms . WAOA 2023. Lecture Notes in Computer Science, vol 14297. Springer, Cham. https://doi.org/10.1007/978-3-031-49815-2_15

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-49815-2_15

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-49814-5

  • Online ISBN: 978-3-031-49815-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation