Graph-TSP from Steiner Cycles

Abstract

AbstractWe 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.KeywordsSpanning TreeHamiltonian PathSimple CycleGraph MetricsBack EdgeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.