ABSTRACT
This chapter discusses the concept of approximation stability and explores the concept of stability to the traveling salesman problem (TSP) and exhibit a partition of the set of general TSP input instances into infinitely many classes, according to their approximability in dependence of a relaxation of the triangle inequality. It explains complement the picture by introducing also a partition into infinitely many approximability classes inside the metric TSP. The chapter deals with a survey of other successful applications of approximation stability and a discussion of the concept. The approach guided by the parameterized triangle inequality led to the development of a few new simple and fast algorithms that improve the best known approximation ratio at least for a part of the sharpened-triangle-inequality case. The design of approximation algorithms has evolved as one of the most successful approaches for dealing with hard optimization problems.
