Towards New Bounds for the 2-Edge Connected Spanning Subgraph Problem

Given a complete graph Kn = (V,E) with non-negative edge costs c ∈ R , the problem multi-2ECcost is that of finding a 2-edge connected spanning multi-subgraph of Kn 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 cost of multi-2ECcost, thus become vital to obtain solutions guaranteed to be close to the optimal in a fast manner. In this thesis, we focus on the integrality gap αmulti-2ECcost of multi-2EC LP cost, which is a measure of the quality of multi-2EC cost as a lower bound. Although we currently only know that 6 5 ≤ αmulti-2ECcost ≤ 3 2 , the integrality gap for multi-2ECcost has been conjectured to be 6 5 . We explore the idea of using the structure of solutions for αmulti-2ECcost and the concept of convex combination to obtain improved bounds for αmulti-2ECcost. We focus our efforts on a family J of half-integer solutions that appear to give the largest integrality gap for multi-2ECcost. We successfully show that the conjecture αmulti-2ECcost = 6 5 is true for any cost functions optimized by some x ∈ J . We also study the related problem 2ECsize, 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 2ECsize 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. ii