Approximating generalizations of Max Cut

This thesis is written for the Swedish degree Licentiate of Science, teknisk licentiat. It is a university degree, intermediate between that of master and that of doctor. We study the approximability of different generalizations of the Max Cut problem. First, we show that Max Set Splitting and Max Not-All-Equal Sat are both approximable within 1.380 in probabilistic polynomial time. The algorithm uses standard semidefinite relaxations, combined with a probabilistic post-processing step. Then, we focus on systems of linear equations modulo p with exactly k unknowns in each equations. The naive randomized algorithm, which guesses a solution uniformly at random from the solution space, has performance ratio p for this problem. For k ≥ 3, it has been shown by H̊astad that it is, for all ε > 0, NP-hard to approximate the optimum within p− ε. In contrast to this remarkable result, we show that, for the case when k = 2, it is possible to use a semidefinite relaxation combined with randomized rounding to obtain a probabilistic polynomial time algorithm with performance ratio p− κ(p), where κ(p) > 0 for all p. Finally, we show that it is possible to construct a randomized polynomial time approximation scheme for instances where the number of equations is Θ(n), where n is the number of variables in the instance. TRITA-NA-9809 • ISSN 0348-2952 • ISRN KTH/NA/R--98/9--SE