Department of Applied Mathematics & Physics, Kyoto University

Technical Report 2012-009 (December 4, 2012)

An Exact Algorithm for TSP in Degree-3 Graphs via Circuit Procedure and Amortization on Connectivity Structure
by Mingyu Xiao and Hiroshi Nagamochi

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The paper presents an $O^*(1.2312^n)$-time and polynomial- space algorithm for the traveling salesman problem in an $n$-vertex graph with maximum degree $3$. This improves the previous time bounds of $O^*(1.251^n)$ by Iwama and Nakashima and $O^*(1.260^n)$ by Eppstein. Our algorithm is a simple branch-and-search algorithm. The only branch rule is designed on a cut-circuit structure of a graph induced by unprocessed edges. To improve a time bound by a simple analysis on measure and conquer, we introduce an amortization scheme over the cut-circuit structure by defining the measure of an instance to be the sum of not only weights of vertices but also weights of connected components of the induced graph.