Some New Randomized Approximation Algorithms

The topic of this thesis is approximation algorithms for optimization versions of NP-complete decision problems. No exact algorithms with sub-exponential running times are known for these problems, and therefore approximation algorithms with polynomial running times are studied. An approximation algorithm does not necessarily nd the optimal solution, but it leaves a guarantee of how far from the optimum the output solution can be in the worst case. This performance guarantee is the measure of quality of an approximation algorithm; it should be as close to 1 as possible. We present new approximation algorithms for several di erent maximization problems. All problems are essentially constraint satisfaction problems: An instance consists of a set of constraints on groups of variables. The objective is to satisfy as many of the constraints as possible. Most results on such problems are for binary variables; we give some results for binary variables and some where the domain is Zp. A common feature of all such problems is that they can be approximated within a constant factor by picking a variable assignment uniformly at random. Until recently, this was the best known approximation algorithm for many constraint satisfaction problems. Algorithms based on semide nite programming were introduced by Goemans and Williamson in 1994, and they revolutionized the eld. We continue this line of research and use semide nite programming combined with randomized rounding schemes to obtain algorithms better than picking a solution at random for several di erent problems: Max Set Splitting, Max 3-Horn Sat, Max E2-Lin mod p, and Max p-Section. When restricted to dense instances, most such problems become easier to approximate. We devise a polynomial time approximation scheme for the family Max Ek-Function Sat mod p of constraint satisfaction problems for which the domain is Zp. We also prove lower bounds on the approximability of Max k-Horn Sat and Max E2-Lin mod p. A lower bound in this context is a proof that it is impossible to approximate a problem within some given performance guarantee unless P = NP.