An Improved-time Polynomial-space Exact Algorithm for TSP in Degree-5 Graphs

Abstract

The Traveling Salesman Problem (TSP) is one of the most well-known NP-hard optimization problems. Following a recent trend of research which focuses on developing algorithms for special types of TSP instances, namely graphs of limited degree, in an attempt to reduce a part of the time and space complexity, we present a polynomial-space branching algorithm for the TSP in an n-vertex graph with degree at most 5, and show that it has a running time of O∗(2.3500n), which improves the previous best known time bound of O∗(2.4723n) given by the authors (the 12th International Symposium on Operations Research and Its Application (ISORA 2015), pp.45–58, 2015). While the base of the exponent in the running time bound of our algorithm is greater than 2, it still outperforms Gurevich and Shelah’s O∗(4nnlog n) polynomial-space exact algorithm for the TSP in general graphs (SIAM Journal of Computation, Vol.16, No.3, pp.486–502, 1987). In the analysis of the running time, we use the measure-and-conquer method, and we develop a set of branching rules which foster the analysis of the running time.