Complexity Indices for the Travelling Salesman Problem and Data Mining

We extend our previous work on complexity indices for the travelling salesman problem (TSP) using graph spectral techniques of data mining. A complexity index is an invariant of an instance I by which we can predict the execution time of an exact algorithm for TSP for I. We consider the symmetric travelling salesman problem with instances I represented by complete weighted graphs G. Intuitively, the hardness of an instance G depends on the distribution of short edges within G. Therefore we consider some short edge subgraphs of G (minimal spanning tree and several others) as non-weighted graphs and several their invariants as potential complexity indices. Here spectral invariants (e.g. spectral radius of the adjacency matrix) play an important role. Spectral clustering algorithms are used including information obtained from the spectral gap in Laplacian spectra of short edge subgraphs.