The Raymond and Beverly Sackler Faculty of Exact Sciences School of Mathematical Sciences Approximation Algorithms for NP - Hard Problems in Combinatorial Optimization

This thesis broadly focuses on the design and analysis of algorithmic tools leading to approximation algorithms with provably good performance guarantees. We were especially interested in problems that have highly practical importance yet possess a fundamental structure, such as integer covering, network design, graph partitioning, and discrete location problems. In particular, our main contributions revolve around algorithmic ideas and proof methods that are based on mathematical programming, polyhedral combinatorics, duality, and randomization. The highlights of this work can be briefly described as follows: 1. We devise the first polylogarithmic approximation for the generalized connectivity problem. 2. We present the first non-trivial approximation for the k-Steiner forest problem. 3. We propose the first non-trivial approximation for the k-generalized connectivity problem. 4. We improve on the currently best approximation for directed Steiner network. 5. We present a unified framework for approximating partial covering problems, and demonstrate the applicability of our method in diverse settings.