Abstract
Goemans and Williamson obtained an approximation algorithm for the MAX CUT problem with a performance ratio of α GW ≃ 0.87856. Their algorithm starts by solving a standard SDP relaxation of MAX CUT and then rounds the optimal solution obtained using a random hyperplane. In some cases, the optimal solution of the SDP relaxation happens to lie in a low dimensional space. Can an improved performance ratio be obtained for such instances? We show that the answer is yes in dimensions two and three and conjecture that this is also the case in any higher fixed dimension. In two dimensions an optimal \(\frac{32}{25+5\sqrt{5}}\)-approximation algorithm was already obtained by Goemans. (Note that \(\frac{32}{25+5\sqrt{5}} \simeq 0.88456\).) We obtain an alternative derivation of this result using Gegenbauer polynomials. Our main result is an improved rounding procedure for SDP solutions that lie in ℝ3 with a performance ratio of about 0.8818 . The rounding procedure uses an interesting yin-yan coloring of the three dimensional sphere. The improved performance ratio obtained resolves, in the negative, an open problem posed by Feige and Schechtman [STOC’01]. They asked whether there are MAX CUT instances with integrality ratios arbitrarily close to α GW ≃ 0.87856 that have optimal embedding, i.e., optimal solutions of their SDP relaxations, that lie in ℝ3.
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References
Alon, N., Naor, A.: Approximating the cut-norm via Grothendieck’s inequality. In: Proceedings of the 36th Annual ACM Symposium on Theory of Computing, Chicago, Illinois, pp. 72–80 (2004)
Charikar, M., Wirth, A.: Maximizing Quadratic Programs: Extending Grothendieck’s Inequality. In: Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science, Rome, Italy, pp. 54–60 (2004)
Delorme, C., Poljak, S.: Combinatorial properties and the complexity of a max-cut approximation. European Journal of Combinatorics 14(4), 313–333 (1993)
Delorme, C., Poljak, S.: Laplacian eigenvalues and the maximum cut problem. Mathematical Programming 62(3, Ser. A), 557–574 (1993)
Feige, U., Goemans, M.X.: Approximating the value of two prover proof systems, with applications to MAX-2SAT and MAX-DICUT. In: Proceedings of the 3rd Israel Symposium on Theory and Computing Systems, Tel Aviv, Israel, pp. 182–189 (1995)
Feige, U., Langberg, M.: The RPR2 rounding technique for semidefinite programs. In: Proceedings of the 28th Int. Coll. on Automata, Languages and Programming, Crete, Greece, pp. 213–224 (2001)
Feige, U., Schechtman, G.: On the integrality ratio of semidefinite relaxations of MAX CUT. In: Proceedings of the 33th Annual ACM Symposium on Theory of Computing, Crete, Greece, pp. 433–442 (2001)
Goemans, M.X., Williamson, D.P.: Improved Approximation Algorithms for Maximum Cut and Satisfiability Problems Using Semidefinite Programming. Journal of the ACM 42, 1115–1145 (1995)
Håastad, J.: Some optimal inapproximability results. Journal of the ACM 48(4), 798–859 (2001)
Khot, S., Kindler, G., Mossel, E., O’Donnell, R.: Optimal inapproimability resutls for MAX-CUT and other 2-variable CSPs? In: Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science, Rome, Italy, pp. 146–154 (2004)
Lewin, M., Livnat, D., Zwick, U.: Improved rounding techniques for the MAX 2-SAT and MAX DI-CUT problems. In: Cook, W.J., Schulz, A.S. (eds.) IPCO 2002. LNCS, vol. 2337, pp. 67–82. Springer, Heidelberg (2002)
Matuura, S., Matsui, T.: 0.863-approximation algorithm for MAX DICUT. In: Goemans, M.X., Jansen, K., Rolim, J.D.P., Trevisan, L. (eds.) RANDOM 2001 and APPROX 2001. LNCS, vol. 2129, pp. 138–146. Springer, Heidelberg (2001)
Matuura, S., Matsui, T.: 0.935-approximation randomized algorithm for MAX 2SAT and its derandomization. Technical Report METR 2001-03, Department of Mathematical Engineering and Information Physics, the University of Tokyo, Japan (September 2001)
Schoenberg, I.J.: Positive definite functions on spheres. Duke Math J. 9, 96–107 (1942)
Zwick, U.: Outward rotations: a tool for rounding solutions of semidefinite programming relaxations, with applications to MAX CUT and other problems. In: Proceedings of the 31th Annual ACM Symposium on Theory of Computing, Atlanta, Georgia, pp. 679–687 (1999)
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Avidor, A., Zwick, U. (2005). Rounding Two and Three Dimensional Solutions of the SDP Relaxation of MAX CUT. In: Chekuri, C., Jansen, K., Rolim, J.D.P., Trevisan, L. (eds) Approximation, Randomization and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2005 2005. Lecture Notes in Computer Science, vol 3624. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11538462_2
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DOI: https://doi.org/10.1007/11538462_2
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