Analysis of Detour Gap Numbers

Spanner graphs appear in approximation schemes for problems such as the Traveling Salesman Problem, where an edge weighted graph defines a metric on its vertex set. In such schemes, a spanner is a subgraph of the input graph, which still represents nearly the same metric. We bound its total edge weight using the "detour gap number", which is defined by a linear program. In this thesis, we simplify this linear program, and state the complementary slackness conditions relating it and its dual. We also give a way to prune a graph based on those properties and a counter example showing the detour gap number is not monotone under edge deletion. The thesis also introduced a software package to facilitate the calculation of detour gap numbers.