Randomized Approximation Algorithms: Facility Location, Phylogenetic Networka, Nash Equilibria

Despite a great effort, researchers are unable to find efficient algorithms for a number of natural computational problems. Typically, it is possible to emphasize the hardness of such problems by proving that they are at least as hard as a number of other problems. In the language of computational complexity it means proving that the problem is complete for a certain class of problems. For optimization problems, we may consider to relax the requirement of the outcome to be optimal and accept an approximate (i.e., close to optimal) solution. For many of the problems that are hard to solve optimally, it is actually possible to efficiently find close to optimal solutions. In this thesis, we study algorithms for computing such approximate solutions.

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