A graph is said to be 2-edge-connected if it remains connected after the deletion of any single edge. Given an unweighted bridgeless graph G with n vertices, the minimum size 2-edge-connected spanning subgraph problem (2EC) is that of finding a 2-edge-connected spanning subgraph of G with the minimum number of edges. This problem has important applications in the design of survivable networks. However, because the problem is NP-hard, it is unlikely that efficient methods exist for solving it. Thus efficient methods that find solutions that are provably close to optimal are sought. In this thesis, an approximation algorithm is presented for 2EC on bridgeless cubic graphs which guarantees to be within 5/4 of the optimal solution value, improving on the previous best proven approximation guarantee of 5/4+ε for this problem. We also focus on the linear programming (LP) relaxation of 2EC, which provides important lower bounds for 2EC in useful solution techniques like branch and bound. The “goodness” of this lower bound is measured by the integrality gap of the LP relaxation for 2EC, denoted by α2EC. Through a computational study, we find the exact value of α2EC for graphs with small n. Moreover, a significant improvement is found for the lower bound on the value of α2EC for bridgeless subcubic graphs, which improves the known best lower bound on α2EC from 9/8 to 8/7.

A graph is said to be 2-edge-connected if it remains connected after the deletion of any single edge. Given an unweighted bridgeless graph G with n vertices, the minimum size 2-edge-connected spanning subgraph problem (2EC) is that of finding a 2-edge-connected spanning subgraph of G with the minimum number of edges. This problem has important applications in the design of survivable networks. However, because the problem is NP-hard, it is unlikely that efficient methods exist for solving it. Thus efficient methods that find solutions that are provably close to optimal are sought. In this thesis, an approximation algorithm is presented for 2EC on bridgeless cubic graphs which guarantees to be within 5/4 of the optimal solution value, improving on the previous best proven approximation guarantee of 5/4+ε for this problem. We also focus on the linear programming (LP) relaxation of 2EC, which provides important lower bounds for 2EC in useful solution techniques like branch and bound. The “goodness” of this lower bound is measured by the integrality gap of the LP relaxation for 2EC, denoted by α2EC. Through a computational study, we find the exact value of α2EC for graphs with small n. Moreover, a significant improvement is found for the lower bound on the value of α2EC for bridgeless subcubic graphs, which improves the known best lower bound on α2EC from 9/8 to 8/7.

Given a complete graph K_n = (V,E) with non-negative edge costs c ∈ R^E, the problem multi-2EC_cost is that of finding a 2-edge connected spanning multi-subgraph of K_n with minimum cost. It is believed that there are no efficient ways to solve the problem exactly, as it is NP-hard. Methods such as approximation algorithms, which rely on lower bounds like the linear programming relaxation multi-2EC^LP of multi-2EC , thus become vital cost cost to obtain solutions guaranteed to be close to the optimal in a fast manner. In this thesis, we focus on the integrality gap αmulti-2EC of multi-2EC^LP , which is a measure of the quality of multi-2EC^LP as a lower bound. Although we currently only know cost that 6/5 ≤ αmulti-2EC_cost ≤ 3 , the integrality gap for multi-2EC_cost has been conjectured to be 6/5. We explore the idea of using the structure of solutions for αmulti-2EC_cost and the concept of convex combination to obtain improved bounds for αmulti-2EC_cost. We focus our efforts on a family J of half-integer solutions that appear to give the largest integrality gap for multi-2EC_cost. We successfully show that the conjecture αmulti-2EC_cost = 6/5 is true for any cost functions optimized by some x∗ ∈ J. We also study the related problem 2EC_size, which consists of finding the minimum size 2-edge connected spanning subgraph of a 2-edge connected graph. The problem is NP-hard even at its simplest, when restricted to cubic 3-edge connected graphs. We study that case in the hope of finding a more general method, and we show that every 3-edge connected cubic graph G = (V ′, E′), with n = |V ′| allows a 2EC_size solution for G of size at most 7n/6 This improves upon Boyd, Iwata and Takazawa’s guarantee of 6n/5 and extend Takazawa’s 7n/6 guarantee for bipartite cubic 3-edge connected graphs to all cubic 3-edge connected graphs.

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.

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.