Abstract
We present an approach for the traveling salesman problem with graph metric based on Steiner cycles. A Steiner cycle is a cycle that is required to contain some specified subset of vertices. For a graph \(G\), if we can find a spanning tree \(T\) and a simple cycle that contains the vertices with odd-degree in \(T\), then we show how to combine the classic “double spanning tree” algorithm with Christofides’ algorithm to obtain a TSP tour of length at most \(\frac{4n}{3}\). We use this approach to show that a graph containing a Hamiltonian path has a TSP tour of length at most \(4n/3\).
Since a Hamiltonian path is a spanning tree with two leaves, this motivates the question of whether or not a graph containing a spanning tree with few leaves has a short TSP tour. The recent techniques of Mömke and Svensson imply that a graph containing a depth-first-search tree with \(k\) leaves has a TSP tour of length \(4n/3 + O(k)\). Using our approach, we can show that a \(2(k-1)\)-vertex connected graph that contains a spanning tree with at most \(k\) leaves has a TSP tour of length \(4n/3\). We also explore other conditions under which our approach results in a short tour.
R. Ravi: Supported in part by NSF grants CCF1143998 and CCF1218382.
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Iwata, S., Newman, A., Ravi, R. (2014). Graph-TSP from Steiner Cycles. In: Kratsch, D., Todinca, I. (eds) Graph-Theoretic Concepts in Computer Science. WG 2014. Lecture Notes in Computer Science, vol 8747. Springer, Cham. https://doi.org/10.1007/978-3-319-12340-0_26
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