An Improved Approximation Algorithm for TSP with Distances One and Two

Markus Bläser, L. Shankar Ram

Abstract: The minimum traveling salesman problem with distances one and two is the following problem: Given a complete undirected graph G=(V,E) with a cost function w: E→ {1, 2}, find a Hamiltonian tour of minimum cost. In this paper, we provide an approximation algorithm for this problem achieving a performance guarantee of \(\frac{315}{271}\). This algorithm can be further improved obtaining a performance guarantee of \(\frac{65}{56}\). This is better than the one achieved by Papadimitriou and Yannakakis [8], with a ratio \(\frac{7}{6}\), more than a decade ago. We enhance their algorithm by an involved procedure and find an improved lower bound for the cost of an optimal Hamiltonian tour.