Approximation algorithms for the traveling salesman problem

Jérôme Monnot (1) , Vangelis Th. Paschos (1) , Sophie Toulouse (1)

1 LAMSADE - Laboratoire d'analyse et modélisation de systèmes pour l'aide à la décision

We first prove that the minimum and maximum traveling salesman problems, their metric versions as well as some versions defined on parameterized triangular inequalities (called sharpened and relaxed metric traveling salesman) are all equi-approximable under an approximation measure, called differential-approximation ratio, that measures how the value of an approximate solution is placed in the interval between the worst- and the best-value solutions of an instance. We next show that the 2_OPT, one of the most known traveling salesman algorithms, approximately solves all these problems within differential-approximation ratio bounded above by 1/2. We also provide some differential inapproximation results for the metric traveling salesman and its special case with distances 1 and 2. Finally, we provide some standard approximaiton results for the maximum sharpened and relaxed traveling salesman problems, which (to our knowledge) have not been studied until know.

Mots clés: metric TSP; differential approximation; traveling salesman problem; approximation algorithms

Domaines: Algorithme et structure de données [cs.DS]; Complexité [cs.CC]; Mathématique discrète [cs.DM]

Jérôme Monnot, Vangelis Th. Paschos, Sophie Toulouse. Approximation algorithms for the traveling salesman problem. Mathematical Methods of Operations Research, 2003, 56 (3), pp.387-405. ⟨10.1007/s001860200239⟩. ⟨hal-00017262⟩