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Published online by Cambridge University Press: 14 November 2024
By combining the removable pairing technique presented in Chapter 12 with a new approach based on ear-decompositions and matroid intersection, Sebő and Vygen improved the approximation ratio for Graph TSP from 13/9 to 7/5. We will present this algorithm, which is still the best-known approximation algorithm for Graph TSP, in this chapter.
An interesting feature of this algorithm is that it is purely combinatorial, does not need to solve a linear program, and runs in O(n3) time. To describe the algorithm, we review some matching theory, including a theorem of Frank that links ear-decompositions to T-joins. A slight variant of the Graph TSP algorithm is a 4/3-approximation algorithm for finding a smallest 2-edge-connected spanning subgraph, which was the best known for many years. The proofs will also imply corresponding upper bounds on the integrality ratios.
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