Traveling Salesman Should not be Greedy: Domination Analysis of Greedy-Type Heuristics for the TSP

Abstract

Computational experiments show that the greedy algorithm (GR)
and the nearest neighbor algorithm (NN), popular choices for tour
construction heuristics, work at acceptable level for the Euclidean TSP,
but produce very poor results for the general Symmetric and Asymmetric
TSP (STSP and ATSP). We prove that for every n >= 2 there
is an instance of ATSP (STSP) on n vertices for which GR finds the
worst tour. The same result holds for NN. We also analyze the repetitive
NN (RNN) that starts NN from every vertex and chooses the best
tour obtained. We prove that, for the ATSP, RNN always produces
a tour, which is not worse than at least n/2 − 1 other tours, but for
some instance it finds a tour, which is not worse than at most n − 2
other tours, n >= 4. We also show that, for some instance of the STSP
on n >= 4 vertices, RNN produces a tour not worse than at most 2^(n−3) tours. These results are in sharp contrast to earlier results by G. Gutin and A. Yeo, and A. Punnen and S. Kabadi, who proved that, for the ATSP, there are tour construction heuristics, including some popular ones, that always build a tour not worse than at least (n − 2)! tours.

Keywords: TSP, domination analysis, greedy algorithm, nearest
neighbor algorithm