Approximating Integer Quadratic Programs and MAXCUT in Subdense Graphs

  • Andreas Björklund
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3669)


Let A be a real symmetric n× n-matrix with eigenvalues λ1,⋯,λn ordered after decreasing absolute value, and b an n× 1-vector. We present an algorithm finding approximate solutions to min x*(Ax + b) and max x*(Ax + b) over x∈ {–1,1}n, with an absolute error of at most \((c_{1}|{\rm \lambda_{1}}|+|{\rm \lambda}_{\lceil c_{2} {\rm log}n\rceil}|)2n+O((\alpha n+\beta)\sqrt{n {\rm log}n})\), where α and β are the largest absolute values of the entries in A and b, respectively, for any positive constants c1 and c2, in time polynomial in n.

We demonstrate that the algorithm yields a PTAS for MAXCUT in regular graphs on n vertices of degree d of \(\omega(\sqrt{n{\rm log}n})\), as long as they contain O(d4log n) 4-cycles. The strongest previous result showed that Ω(n/log n) average degree graphs admit a PTAS.

We also show that smooth n-variate polynomial integer programs of constant degree k, always can be approximated in polynomial time leaving an absolute error of o(nk), answering in the affirmative a suspicion of Arora, Karger, and Karpinski in STOC 1995.


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  1. 1.
    Alon, N., de la Vega, W.F., Kannan, R., Karpinski, M.: Random Sampling and Approximation of MAX-CSP Problems. In: Proc. 34th STOC, pp. 534–543. ACM, New York (2002); The full paper can be found in Technical Report TR01-100, ECCC (2001)Google Scholar
  2. 2.
    Arora, S., Berger, E., Hazan, E., Kindler, G., Safra, M.: On Non-Approximability for Quadratic Programs. Technical Report TR05-58, ECCC, 2005. To appear at Proc. 46th FOCS, IEEE (2005)Google Scholar
  3. 3.
    Arora, S., Karger, D., Karpinski, M.: Polynomial time approximation schemes for dense instances of NP-hard problems. In: Proc. 27th STOC, pp. 284–293. ACM, New York (1995)Google Scholar
  4. 4.
    Arora, S., Lund, C., Motwani, R., Sudan, M., Szegedy, M.: Proof verification and hardness of approximation problems. In: Proc. 33rd FOCS, pp. 14–23. IEEE, Los Alamitos (1992)Google Scholar
  5. 5.
    Biggs, N.: Algebraic Graph Theory. Cambridge University Press, Cambridge (1996); ISBN 0-521-45897-8Google Scholar
  6. 6.
    de la Vega, W.F.: MAX-CUT has a Randomized Approximation Scheme in Dense Graphs. Random Structures and Algorithms 8(3), 187–198 (1996)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    de la Vega, W.F., Karpinski, M.: Polynomial time approximation of dense weighted instances of MAX-CUT. Random Structures and Algorithms 16(4), 314–332 (2000)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    de la Vega, W.F., Karpinski, M.: A Polynomial Time Approximation Scheme for Subdense MAX-CUT. Technical Report TR02-044, ECCC (2002)Google Scholar
  9. 9.
    Goemans, M.X., Williamson, D.P.: Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM 42, 1115–1145 (1995)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Goldreich, O., Goldwasser, S., Ron, D.: Property Testing and its Connection to Learning and Approximation. In: Proc. 37th FOCS, vol. 45, pp. 339–348. IEEE, Los Alamitos (1996); The full paper can be found in J. ACM 45, 653–750 (1998)Google Scholar
  11. 11.
    Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. The John Hopkins Universal Press, Baltimore (1996); ISBN 0-8018-5414-8MATHGoogle Scholar
  12. 12.
    Karloff, H.: Linear Programming. Birkhäuser, Boston (1991); ISBN 3-7643-3561-0MATHCrossRefGoogle Scholar
  13. 13.
    Papadimitriou, C.H., Yannakakis, M.: Optimization, approximation, and complexity classes. J. Comput. System Sci. 43, 425–440 (1991)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Raghavan, P.: Probabilistic construction of deterministic algorithms: Approximate packing integer programs. J. Comput. System Sci. 37(2), 130–143 (1988)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Raghavan, P., Thompson, C.: Randomized Rounding: a technique for provably good algorithms and algorithmic proofs. Combinatorica 7, 365–374 (1987)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Andreas Björklund
    • 1
  1. 1.Department of Computer ScienceLund UniversityLundSweden

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