Approximation algorithms for the traveling salesman problem with range condition

We prove that the Christofides algorithm gives a 4/3 approximation ratio for the special case of traveling salesman problem (TSP) in which the maximum weight in the given graph is at most twice the minimum weight for the odd degree restricted graphs. A graph is odd degrees restricted if the number of odd degree vertices in any minimum spanning tree of the given graph is less than 1/4 times the number of vertices in the graph. We Drove that the Christofides algorithm is more efficient tin terms of runtime) than the previous existing algorithms for this special case of the traveling salesman problem. Secondly, we apply the concept. of stability of approximation to this special case of traveling salesman problem in order to partition the set of all instances of TSP into art infinite spectrum of classes according to their approximability. AMS Subject Classification. 68W25, 05C85, 68W40.