A Polynomial-Space Exact Algorithm for TSP in Degree-6 Graphs

Previously, the authors of this work have presented in a series of papers polynomial-space algorithms for the TSP in graphs with degree at most five, six and seven. Each of these algorithms is the first algorithm specialized for the TSP in graphs of limited degree five, six, and seven respectively, and the running time bound of these algorithms outperforms Gurevich and Shelah’s O∗(4nnlogn) algorithm for the TSP in n-vertex graphs (SIAM Journal of Computation, 16(3), pp. 486–502, 1987). Now we ask what is the highest degree i until which a specialized polynomial-space algorithm for the TSP in graphs with maximum degree i outperforms Gurevich and Shelah’s O∗(4nnlogn) algorithm? As an answer to this question, this paper presents the first polynomial-space exact algorithm specialized for the TSP in graphs with degree at most eight. We develop a set of branching rules to aid the analysis of the branching algorithm, and we use the measure-and-conquer method to effectively analyze our branching algorithm. We obtain a running time of O∗(4.1485n), and this running time bound does not give an advantageous algorithm for the TSP in degree-8 graphs over Gurevich and Shelah’s algorithm for the TSP in general, but it gives a limit as to the applicability of our choice of branching rules and analysis method for designing a polynomial-space exact algorithm for the TSP in graphs of limited degree.