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, and thus alleviating a part of the time and space complexity, we
present a polynomial-space branching algorithm for the TSP in graphs
with degree at most~$5$, and show that it has a running time of $O^*(2.
4531^n )$.
To the best of our knowledge, this is the first exact algorithm
specialized to graphs of such high degree.
While the base of the exponent in the running time bound is greater than
two, our algorithm uses space merely polynomial in an input instance
size, and thus by far outperforms Gurevich and Shelah's~$O^*(4^n n^{\log
n})$ polynomial-space exact algorithm for the general TSP (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.