Abstract
We show how to find a Hamiltonian cycle in a graph of degree at most three with n vertices, in time \(\mathcal{O}(2^{n/3}) \approx {\rm 1.25992}^n\) and linear space. Our algorithm can find the minimum weight Hamiltonian cycle (traveling salesman problem), in the same time bound, and count the number of Hamiltonian cycles in time \(\mathcal{O}(2^{3n/8}n^{\mathcal{O}(1)}) \approx {\rm 1.29684}^n\). We also solve the traveling salesman problem in graphs of degree at most four, by a randomized (Monte Carlo) algorithm with runtime \(\mathcal{O}((27/4)^{n/3}) \approx {\rm 1.88988}^n\). Our algorithms allow the input to specify a set of forced edges which must be part of any generated cycle.
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Eppstein, D. (2003). The Traveling Salesman Problem for Cubic Graphs. In: Dehne, F., Sack, JR., Smid, M. (eds) Algorithms and Data Structures. WADS 2003. Lecture Notes in Computer Science, vol 2748. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-45078-8_27
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DOI: https://doi.org/10.1007/978-3-540-45078-8_27
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