In this thesis we focus on two NP-hard and intensively studied problems: The travelling salesman problem (TSP), which aims to find a minimum cost tour that visits every node exactly once in a complete weighted graph, and the 2-edge-connected spanning subgraph problem (2ECSS), which aims to find a minimum size 2-edge-connected spanning subgraph in a given graph. TSP and 2ECSS have many real world applications. However, both problems are NP-hard which means it is unlikely that polynomial time algorithms exist to solve them, so methods that return close to optimal solutions are sought. In this thesis we mainly focus on k-approximation algorithms for the two problems, which efficiently return a solution within k times of the optimal solution. For a special case of TSP called graph TSP, using ideas from Momke and Svensson, we present a 25/18-approximation algorithm for a special class of graphs using circulations and T-joins, which improves the previous known best bound of 7/5 for such graphs. Moreover, if the graph does not contain special nodes, our algorithm ensures the ratio of 4/3. For 2ECSS, given any k-edge-connected graph G=(V,E), |V|=n, |E|=m, we present an approximation algorithm that gives a 2-edge-connected spanning subgraph with the number of edges at most n+(m-n)/(k-1)-(k-2)/(k-1) with a novel use of circulations, which improves both the approximation ratio and the simplicity of the proof compared to a result by Huh in 2004.

The Metric Travelling Salesman Problem, henceforth metric TSP, is a fundamental problem in combinatorial optimization which consists of finding a minimum cost Hamiltonian cycle (also called a TSP tour) in a weighted complete graph in which the costs are metric. Metric TSP is known to belong to a class of problems called NP-hard even in the special case of graph-TSP, where the metric costs are based on a given graph. Thus, it is highly unlikely that efficient methods exist for solving large instances of these problems exactly. In this thesis, we develop a new heuristic for metric TSP based on extending ideas successfully used by Mömke and Svensson for the special case of graph-TSP to the more general case of metric TSP. We demonstrate the efficiency and usefulness of our heuristic through empirical testing. Additionally, we turn our attention to graph-TSP. For this special case of metric TSP, there has been much recent progress with regards to improvements on the cost of the solutions. We find the exact value of the ratio between the cost of the optimal TSP tour and the cost of the optimal subtour linear programming relaxation for small instances of graph-TSP, which was previously unknown. We also provide a simplified algorithm for special graph-TSP instances based on the subtour linear programming relaxation.

In this thesis we focus on two NP-hard and intensively studied problems: The travelling salesman problem (TSP), which aims to find a minimum cost tour that visits every node exactly once in a complete weighted graph, and the 2-edge-connected spanning subgraph problem (2ECSS), which aims to find a minimum size 2-edge-connected spanning subgraph in a given graph. TSP and 2ECSS have many real world applications. However, both problems are NP-hard which means it is unlikely that polynomial time algorithms exist to solve them, so methods that return close to optimal solutions are sought. In this thesis we mainly focus on k-approximation algorithms for the two problems, which efficiently return a solution within k times of the optimal solution. For a special case of TSP called graph TSP, using ideas from Momke and Svensson, we present a 25/18-approximation algorithm for a special class of graphs using circulations and T-joins, which improves the previous known best bound of 7/5 for such graphs. Moreover, if the graph does not contain special nodes, our algorithm ensures the ratio of 4/3. For 2ECSS, given any k-edge-connected graph G=(V,E), |V|=n, |E|=m, we present an approximation algorithm that gives a 2-edge-connected spanning subgraph with the number of edges at most n+(m-n)/(k-1)-(k-2)/(k-1) with a novel use of circulations, which improves both the approximation ratio and the simplicity of the proof compared to a result by Huh in 2004.