Department of Applied Mathematics & Physics, Kyoto University

Technical Report 2015-002 (June 22, 2015)

A Polynomial-space Exact Algorithm for TSP in Degree-5 Graphs
by Norhazwani Md Yunos, Aleksandar Shurbevski, Hiroshi Nagamochi

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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.