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European Symposium on Algorithms

ESA 2006: Algorithms – ESA 2006 pp 432–443Cite as

Approximating Almost All Instances of Max-Cut Within a Ratio Above the Håstad Threshold

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4168))

Abstract

We give a deterministic polynomial-time algorithm which for any given average degree d and asymptotically almost all random graphs G in \(\mathcal G(n, m= \lfloor\frac{d}{2}n\rfloor)\) outputs a cut of G whose ratio (in cardinality) with the maximum cut is at least 0.952. We remind the reader that it is known that unless P=NP, for no constant ε>0 is there a Max-Cut approximation algorithm that for all inputs achieves an approximation ratio of (16/17) +ε (16/17 < 0.94118).

The authors are partially supported by European Social Fund (ESF), Operational Program for Educational and Vocational Training II (EPEAEK II), and particularly Pythagoras. The second author is partially supported by Future and Emerging Technologies programme of the EU under contract 001907 “Dynamically Evolving, Large-Scale Information Systems (DELIS).”

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References

  1. Barahona, F., Grotschel, M., Junger, M., Reinelt, G.: An application of combinatorial optimization to statistical physics and circuit layout design. Operations Research 36, 493–513 (1988)

    Article  MATH  Google Scholar 

  2. Beis, M., Duckworth, W., Zito, M.: Packing edges in random regular graphs. In: Diks, K., Rytter, W. (eds.) MFCS 2002. LNCS, vol. 2420, pp. 118–130. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  3. Beis, M., Duckworth, W., Zito, M.: Large k-separated matchings of random regular graphs. In: Estivill-Castro, V. (ed.) ACSC, CRPIT, vol. 38, pp. 175–182. Australian Computer Society (2005)

    Google Scholar 

  4. Bertoni, A., Campadelli, P., Posenato, R.: Un upper bound for the maximum cut mean value. In: Möhring, R.H. (ed.) WG 1997. LNCS, vol. 1335, pp. 78–84. Springer, Heidelberg (1997)

    Chapter  Google Scholar 

  5. Charikar, M., Makarychev, K., Makarychev, Y.: Near-optimal algorithms for unique games. In: Proceedings of the 38th ACM Symposium on Theory of Computing, Seattle, Washington, USA (2006)

    Google Scholar 

  6. Coja-Oghlan, A., Moore, C., Sanwalani, V.: Max k-cut and approximating the chromatic number of random graphs. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) ICALP 2003. LNCS, vol. 2719, pp. 200–211. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  7. Coppersmith, D., Gamarnik, D., Hajiaghayi, M.T., Sorkin, G.B.: Random max sat, random max cut, and their phase transitions. Random Struct. Algorithms 24(4), 502–545 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  8. de la Vega, W.F., Karpinski, M.: 9/8-approximation algorithm for random max-3sat. Electronic Colloquium on Computational Complexity (ECCC) (070) (2002)

    Google Scholar 

  9. Díaz, J., Grammatikopoulos, G., Kaporis, A.C., Kirousis, L.M., Pérez, X., Sotiropoulos, D.G.: 5-regular graphs are 3-colorable with positive probability. In: Brodal, G.S., Leonardi, S. (eds.) ESA 2005. LNCS, vol. 3669, pp. 215–225. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  10. Dubois, O., Mandler, J.: On the non-3-colourability of random graphs. ArXiv Mathematics e-prints (September 2002)

    Google Scholar 

  11. Frieze, A.M., McDiarmid, C.: Algorithmic theory of random graphs. Random Struct. Algorithms 10(1–2), 5–42 (1997)

    Article  MathSciNet  Google Scholar 

  12. Goemans, M., Williamson, D.: Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. Journal of the ACM 42, 1115–1145 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  13. Håstad, J.: Some optimal inapproximability results. Journal of the ACM 48, 798–869 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  14. Interian, Y.: Approximation algorithm for random MAX-k-SAT. In: Hoos, H., Mitchell, D.G. (eds.) SAT 2004. LNCS, vol. 3542, pp. 173–182. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  15. Kalapala, V., Moore, C.: Max-cut on sparse random graphs. TR-CS-2002-24, University of New Mexico Department of Computer Science (2002)

    Google Scholar 

  16. Khot, S.: Hardness of approximating the shortest vector problem in lattices. In: FOCS, pp. 126–135. IEEE Computer Society, Los Alamitos (2004)

    Google Scholar 

  17. Mitzenmacher, M.: Tight thresholds for the pure literal rule. Technical report, Digital Equipment Corporation (1997), available at: www.research.compaq.com/SRC/

  18. Van Ngoc, N., Tuza, Z.: Linear-time algorithms for the max cut problem. Combinatorics, Probability & Computing 2, 201–210 (1993)

    Article  MATH  Google Scholar 

  19. Poljak, S., Tuza, Z.: The expected relative error of the polyhedral approximation of the max-cut problem. Operations Research Letters 16, 191–198 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  20. Trevisan, L., Sorkin, G., Sudan, M., Williamson, D.: Gadgets, approximation, and linear programming. SIAM Journal on Computing 29(6), 2074–2097 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  21. Wormald, N.C.: Differential equations for random processes and random graphs. The Annals of Applied Probability 5(4), 1217–1235 (1995)

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Computer Engineering and Informatics, University of Patras, GR-265 04, Patras, Greece

    A. C. Kaporis, L. M. Kirousis & E. C. Stavropoulos

  2. Research Academic Computer Technology Institute, P.O. Box 1122, GR-261 10, Patras, Greece

    L. M. Kirousis

Authors
  1. A. C. Kaporis
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  2. L. M. Kirousis
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  3. E. C. Stavropoulos
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Editor information

Editors and Affiliations

  1. Microsoft Research, One Microsoft Way, Redmond, WA 98052, USA and School of Computer Science, Tel-Aviv University, 69978, Tel-Aviv, Israel

    Yossi Azar

  2. Department of Computer Science, University of Leicester, University Road, LE1 7RH, Leicester, UK

    Thomas Erlebach

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© 2006 Springer-Verlag Berlin Heidelberg

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Kaporis, A.C., Kirousis, L.M., Stavropoulos, E.C. (2006). Approximating Almost All Instances of Max-Cut Within a Ratio Above the Håstad Threshold. In: Azar, Y., Erlebach, T. (eds) Algorithms – ESA 2006. ESA 2006. Lecture Notes in Computer Science, vol 4168. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11841036_40

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  • DOI: https://doi.org/10.1007/11841036_40

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-38875-3

  • Online ISBN: 978-3-540-38876-0

  • eBook Packages: Computer ScienceComputer Science (R0)

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