Approximability of TSP on Power Law Graphs

TL;DR

This paper analyzes the approximability of the Traveling Salesman Problem on power law graphs, providing improved algorithms and bounds for specific ranges of the power law exponent, with both deterministic and random graph models.

Contribution

It offers refined approximation algorithms for TSP on power law graphs and establishes the first explicit inapproximability bounds for these cases.

Findings

Achieved a 1.34-approximation ratio for Graphic TSP at β=1.5.

Improved approximation ratios for (1,2)-TSP on deterministic PLGs for β > 1.666.

Derived better expected approximation ratios for random PLGs with β between 1 and 3.5.

Abstract

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.