Hicks, Illya V.2014-08-262014-08-262014-052014-04-11May 2014Fast, Caleb. "On the Integrality Gap of the Subtour Relaxation of the Traveling Salesman Problem for Certain Fractional 2-matching Costs." (2014) Master’s Thesis, Rice University. <a href="https://hdl.handle.net/1911/76721">https://hdl.handle.net/1911/76721</a>.https://hdl.handle.net/1911/76721This thesis provides new bounds on the strength of the subtour relaxation of the Traveling Salesman Problem (TSP) for fractional 2-matching cost instances whose support graphs have no fractional cycles larger than five vertices. This work provides insight for improving approximation heuristics for the TSP and into the structure of solutions produced by the subtour relaxation. Guided by a T-join derived from the subtour relaxation, I form a tour by adding edges to the subtour relaxation. By this constructive process, I prove that the optimal solution of the TSP is within 4/3, 17/12, or strictly less than 3/2 of the optimal solution of the subtour relaxation. Thus, this thesis takes a step towards proving the 4/3 conjecture for the TSP and the development of a 4/3 approximation algorithm for the TSP. These developments would provide improved approximations for applications such as DNA sequencing, route planning, and circuit board testing.application/pdfengCopyright is held by the author, unless otherwise indicated. Permission to reuse, publish, or reproduce the work beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder.Traveling salesman problem (TSP)Fractional 2-matching instanceSubtour elimination4/3 conjectureIntegrality gapApplied mathematicsThesis2014-08-26