Approximation Algorithms for the Asymmetric Traveling Salesman Problem : Describing two recent methods

TL;DR

This paper reviews two recent approximation algorithms for the asymmetric traveling salesman problem, highlighting their methods, improvements over previous work, and their approximation ratios.

Contribution

It provides an intuitive description of the algorithms by Feige-Singh and Asadpour et al., and discusses their improvements and significance in approximating ATSP.

Findings

Feige-Singh improved the approximation ratio to 0.66 from 0.84.

Combined results establish an approximation ratio of (4/3 + ε) log n for ATSPP.

Asadpour et al. introduced a randomized algorithm with ratio O(log n / log log n).

Abstract

The paper provides a description of the two recent approximation algorithms for the Asymmetric Traveling Salesman Problem, giving the intuitive description of the works of Feige-Singh[1] and Asadpour et.al\ [2].\newline [1] improves the previous $O(\log n)$ approximation algorithm, by improving the constant from 0.84 to 0.66 and modifying the work of Kaplan et. al\ [3] and also shows an efficient reduction from ATSPP to ATSP. Combining both the results, they finally establish an approximation ratio of $\left(\frac{4}{3}+\epsilon \right)\log n$ for ATSPP,\ considering a small $\epsilon>0$,\ improving the work of Chekuri and Pal.[4]\newline Asadpour et.al, in their seminal work\ [2], gives an $O\left(\frac{\log n}{\log \log n}\right)$ randomized algorithm for the ATSP, by symmetrizing and modifying the solution of the Held-Karp relaxation problem and then proving an exponential family distribution for probabilistically constructing a maximum entropy spanning tree from a spanning tree polytope and then finally defining the thin-ness property and transforming a thin spanning tree into an Eulerian walk.\ The optimization methods used in\ [2] are quite elegant and the approximation ratio could further be improved, by manipulating the thin-ness of the cuts.