The greedy and nearest-neighbor TSP heuristics can both have $\log n$ approximation factors from optimal in worst case, even just for $n$ points in Euclidean space. In this note, we show that this approximation factor is only realized when the optimal tour is unusually short. In particular, for points from any fixed $d$-Ahlfor's regular metric space (which includes any $d$-manifold like the $d$-cube $[0,1]^d$ in the case $d$ is an integer but also fractals of dimension $d$ when $d$ is real-valued), our results imply that the greedy and nearest-neighbor heuristics have additive errors from optimal on the order of the optimal tour length through random points in the same space, for $d>1$.
@article{10_37236_12651,
author = {Alan Frieze and Wesley Pegden},
title = {The bright side of simple heuristics for the {TSP}},
journal = {The electronic journal of combinatorics},
year = {2024},
volume = {31},
number = {4},
doi = {10.37236/12651},
zbl = {1558.90140},
url = {http://geodesic.mathdoc.fr/articles/10.37236/12651/}
}
TY - JOUR
AU - Alan Frieze
AU - Wesley Pegden
TI - The bright side of simple heuristics for the TSP
JO - The electronic journal of combinatorics
PY - 2024
VL - 31
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.37236/12651/
DO - 10.37236/12651
ID - 10_37236_12651
ER -
%0 Journal Article
%A Alan Frieze
%A Wesley Pegden
%T The bright side of simple heuristics for the TSP
%J The electronic journal of combinatorics
%D 2024
%V 31
%N 4
%U http://geodesic.mathdoc.fr/articles/10.37236/12651/
%R 10.37236/12651
%F 10_37236_12651
Alan Frieze; Wesley Pegden. The bright side of simple heuristics for the TSP. The electronic journal of combinatorics, Tome 31 (2024) no. 4. doi: 10.37236/12651
