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Published online by Cambridge University Press: 14 November 2024
After the O(log n)-approximation algorithms for Asymmetric TSP, the first algorithm to beat the cycle cover algorithm by more than a constant factor was found in 2009 by Asadpour, Goemans, Mądry, Oveis Gharan, and Saberi. Their approach is based on finding a "thin" (oriented) spanning tree and then adding edges to obtain a tour. A major open question is how thin trees are guaranteed to exist.
The O(log n/loglog n)-approximation algorithm by Asadpour et al. samples a random spanning tree from the maximum entropy distribution. To show how this works, we discuss interesting connections between random spanning trees and electrical networks. Some results of this chapter will be used again in Chapters 10 and 11.
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