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.
| Identifer | oai:union.ndltd.org:uottawa.ca/oai:ruor.uottawa.ca:10393/36202 |
| Date | January 2017 |
| Creators | Legault, Philippe |
| Contributors | Boyd, Sylvia |
| Publisher | Université d'Ottawa / University of Ottawa |
| Source Sets | Université d’Ottawa |
| Language | English |
| Detected Language | English |
| Type | Thesis |
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