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Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. (English) Zbl 0885.68088
Summary: We present randomized approximation algorithms for the maximum cut (MAX CUT) and maximum 2-satisfiability (MAX 2SAT) problems that always deliver solutions of expected value at least .87856 times the optimal value. These algorithms use a simple and elegant technique that randomly rounds the solution to a nonlinear programming relaxation. This relaxation can be interpreted both as a semidefinite program and as an eigenvalue minimization problem. The best previously known approximation algorithms for these problems had performance guarantees of $\frac{1}{2}$ for MAX CUT and $\frac{3}{4}$ for MAX 2SAT. Slight extensions of our analysis lead to a .79607-approximation algorithm for the maximum directed cut problem (MAX DICUT) and a .758-approximation algorithm for MAX SAT, where the best previously known approximation algorithms had performance guarantees of $\frac{1}{4}$ and $\frac{3}{4}$, respectively. Our algorithm gives the first substantial progress in approximating MAX CUT in nearly twenty years, and represents the first use of semidefinite programming in the design of approximation algorithms.

##### MSC:
 68Q25 Analysis of algorithms and problem complexity 90C27 Combinatorial optimization 90C35 Programming involving graphs or networks 90C59 Approximation methods and heuristics
##### Keywords:
randomized approximation algorithms
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