Survivable network design is an important area of combinatorial optimization. In a classical setting, we are given a network represented as a graph, and the goal is to select the cheapest subset of edges that guarantee some connectivity requirements among its nodes, even if some of its elements (like nodes or edges) might fail. Survivable Network Design problems have various applications in Transportation Systems and Wireless Network Services. For instance,  one can model transportation systems as graphs in which the cities play the role of vertices and the roads or railways are the edges of the graph. An example of edge failure in such system is when roads get blocked due to traffic jam, or car accidents. 

In this talk, we consider two classical and well-studied variant of survivable network design problems and provide improve approximations for two families of these problem. 

In the first variant of the problem known as kECCS (or kVCSS), the goal is to find a minimum size k-edge-connected (or k-vertex-connected) spanning subgraph of a given graph. In the second variant known as kECAP (or kVCAP), we are given a k-edge-connected graph G (or k-vertex-connected) graph and a set of additional edges, known as links and the goal is to add a minimum-size subset of links to G such that the edge-connectivity (or vertex-connectivity) of G increases to k+1.