Bounded Degree Graphs Computed for Traveling Salesman Problem Based on Frequency Quadrilaterals

Abstract

Traveling salesman problem (TSP) is well-known in combinatorial optimization. Recent research demonstrates that there are competitive algorithms for TSP on the bounded degree or genus graphs. We present an iterative algorithm to convert the complete graph of a TSP instance into a bounded degree graph based on frequency quadrilaterals. First, the frequency of each edge is computed with N frequency quadrilaterals. Second, \(\frac{1}{3}\) of the edges having the smallest frequencies are cut for each vertex. The two steps are repeated until there are no quadrilaterals for most edges in the residual graph. At the \(k^{th}\ge 1\) iteration, the maximum vertex degree is smaller than \(\left\lceil \left( \frac{2}{3}\right) ^k(n-1)\right\rceil \). Two theorems illustrate that the long edges are cut and the short edges will be preserved. Thus, the optimal solution or a good approximation will be found in the bounded degree graphs. The iterative algorithm is tested with the real-life TSP instances.