 
Summary: Lower Bounds for Insertion Methods for TSP
Yossi Azar \Lambda
Abstract
We show that the random insertion method for the traveling salesman prob
lem (TSP) may produce a tour \Omega\Gammaur/ log n= log log log n) times longer than the
optimal tour. The lower bound holds even in the Euclidean Plane. This is
in contrast to the fact that the random insertion method performs extremely
well in practice. In passing we show that other insertion methods may produce
tours \Omega\Gammaurs n= log log n) times longer than the optimal one. No nonconstant
lower bounds were previously known.
\Lambda Department of Computer Science, TelAviv University, Israel.
1 Introduction
The traveling salesman problem (TSP) is one of the most notorious NPhard prob
lems [GJ]. For the special case that distances satisfy the triangle inequality, many
approximation algorithms have been developed and analyzed. The approximation
factor of such an algorithm is the ratio between the length of the tour obtained by
the algorithm and the optimal tour. The relative performance of different heuris
tics is measured by comparing their approximation factors and their running times.
Rosenkrantz et al [RSL] defined and analyzed several heuristics. Insertion methods
