Shortest path queries, graph partitioning and covering problems in worst and beyond worst case settings

In this thesis, we design algorithms for several NP-hard problems in both worst and beyond worst case settings. In the first part of the thesis, we apply the traditional worst case methodology and design approximation algorithms for the Hub Labeling problem; Hub Labeling is a preprocessing technique introduced to speed up shortest path queries. Before this work, Hub Labeling had been extensively studied mainly in the beyond worst case analysis setting, and in particular on graphs with low highway dimension. In this work, we significantly improve our theoretical understanding of the problem and design (worst-case) algorithms for various classes of graphs, such as general graphs, graphs with unique shortest paths and trees, as well as provide matching inapproximability lower bounds for the problem in its most general settings. Finally, we demonstrate a connection between computing a Hub Labeling on a tree and searching for a node in a tree. In the second part of the thesis, we turn to beyond worst case analysis and extensively study the stability model introduced by Bilu and Linial in an attempt to describe real-life instances of graph partitioning and clustering problems. Informally, an instance of a combinatorial optimization problem is stable if it has a unique optimal solution that remains the unique optimum under small multiplicative perturbations of the parameters of the input. Utilizing the power of convex relaxations for stable instances, we obtain several results for problems such as Edge/Node Multiway Cut, Independent Set (and its equivalent, in terms of exact solvability, Vertex Cover), clustering problems such as $k$-center and $k$-median and the symmetric Traveling Salesman problem. We also provide strong lower bounds for certain families of algorithms for covering problems, thus exhibiting potential barriers towards the design of improved algorithms in this framework.