Improved approximation algorithms for MAXk-CUT and MAX BISECTION

Polynomial-time approximation algorithms with nontrivial performance guarantees are presented for the problems of (a) partitioning the vertices of a weighted graph intok blocks so as to maximize the weight of crossing edges, and (b) partitioning the vertices of a weighted graph into two blocks of equal cardinality, again so as to maximize the weight of crossing edges. The approach, pioneered by Goemans and Williamson, is via a semidefinite programming relaxation.

[1]  Mihalis Yannakakis,et al.  Optimization, approximation, and complexity classes , 1991, STOC '88.

[2]  E. Bofinger,et al.  THE CORRELATION OF MAXIMA IN SAMPLES DRAWN FROM A BIVARIATE NORMAL DISTRIBUTION1 , 1965 .

[3]  L. Wolsey.  Heuristic analysis, linear programming and branch and bound , 1980 .

[4]  Marek Karpinski,et al.  Polynomial time approximation schemes for dense instances of NP-hard problems , 1995, STOC '95.

[5]  G. N. Watson,et al.  Notes on Generating Functions of Polynomials: (2) Hermite Polynomials , 1933 .

[6]  Sanjeev Khanna,et al.  On the Hardness of Approximating Max k-Cut and its Dual , 1997, Chic. J. Theor. Comput. Sci..

[7]  Farid Alizadeh,et al.  Interior Point Methods in Semidefinite Programming with Applications to Combinatorial Optimization , 1995, SIAM J. Optim..

[8]  E. C. Titchmarsh,et al.  The theory of functions , 1933 .

[9]  Carsten Lund,et al.  Proof verification and hardness of approximation problems , 1992, Proceedings., 33rd Annual Symposium on Foundations of Computer Science.

[10]  David P. Williamson,et al.  .879-approximation algorithms for MAX CUT and MAX 2SAT , 1994, STOC '94.

[11]  D. Welsh.  Complexity: Knots, Colourings and Counting: Link polynomials and the Tait conjectures , 1993 .

[12]  W. Vega.  MAX-CUT has a randomized approximation scheme in dense graphs , 1996, Random Struct. Algorithms.

[13]  David R. Karger,et al.  Approximate graph coloring by semidefinite programming , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[14]  J. D. T. Oliveira,et al.  The Asymptotic Theory of Extreme Order Statistics , 1979 .