Approximability of TSP on Power Law Graphs

In this paper we study the special case of Graphic TSP where the underlying graph is a power law graph (PLG). We give a refined analysis of some of the current best approximation algorithms and show that an improved approximation ratio can be achieved for certain ranges of the power law exponent $\beta$. For the value of power law exponent $\beta=1.5$ we obtain an approximation ratio of $1.34$ for Graphic TSP. Moreover we study the $(1,2)$-TSP with the underlying graph of $1$-edges being a PLG. We show improved approximation ratios in the case of underlying deterministic PLGs for $\beta$ greater than $1.666$. For underlying random PLGs we further improve the analysis and show even better expected approximation ratio for the range of $\beta$ between $1$ and $3.5$. On the other hand we prove the first explicit inapproximability bounds for $(1,2)$-TSP for an underlying power law graph.