Non-oblivious 2-opt heuristics for the Traveling Salesman Problem

Worst case analysis of local search based heuristics
Uri Yovel – PhD candidate
Advisor: Asaf Levin
Faculty of Industrial Engineering and Management, Technion
The traveling salesman problem (TSP) and the vehicle routing problem (VRP) are
classical NP-hard problems in combinatorial optimization. The k-opt heuristics are
among the most common techniques for approaching TSP. They are used either
directly or as subroutines in more sophisticated heuristics, such as the celebrated LinKernighan heuristic. The value of k is typically 2 or 3. In this paper, we modify the 2opt heuristic to be based on a function f of the distances rather than the distances
solely. This may be viewed as modifying the local search with the 2-change
neighborhood to be non-oblivious. We denote the corresponding heuristic by (2,f)-opt.
We provide theoretical performance guarantees for it: both lower and upper bounds
based on the ones given by Chandra, Karloff, and Tovey (1999), obtained originally
for the standard 2-opt heuristic; the upper bound is improved by a factor of √2 with
respect to the known upper bound of the standard 2-opt. We then provide
experimental evidence based on TSPLIB benchmark problems, showing that (2,f)-opt
with f(x) = xr for various values of r < 1 significantly outperforms 2-opt.
We then briefly discuss a local search algorithm for the VRP; in particular, we show
that the k-opt heuristic is not strong enough to guarantee a constant approximation
ratio, and we present a different (non-oblivious) local search whose approximation
ratio is 2. To the best of our knowledge, this is the first local search algorithm for this
problem with proved performance guarantee.