Continuous reformulations and heuristics for the Euclidean travelling salesperson problem
ESAIM: COCV
CONTINUOUS REFORMULATIONS AND HEURISTICS FOR THE EUCLIDEAN TRAVELLING SALESPERSON PROBLEM
Tuomo Valkonen 0
Tommi Ka¨rkka¨inen 0
0 Department of Mathematical Information Technology, University of Jyv ̈askyl ̈a , Jyv ̈askyl ̈a , Finland
We consider continuous reformulations of the Euclidean travelling salesperson problem (TSP), based on certain clustering problem formulations. These reformulations allow us to apply a generalisation with perturbations of the Weiszfeld algorithm in an attempt to find local approximate solutions to the Euclidean TSP. Mathematics Subject Classification. 90C26, 90C59, 90C27.
and phrases; Euclidean TSP; clustering; diff-convex; Weiszfeld algorithm
1. Introduction
This paper is concerned with the travelling salesperson problem with Euclidean ( 2) distances (undiscretised),
i.e. the problem of finding the shortest closed path that visits every vertex (or city) in a given finite subset of Rm
exactly once, with the distances given by the Euclidean metric. Whereas various rather efficient algorithms exist
for the general and general metric TSP [
14
], few seem to be able to take advantage of the special features of the
variant with Euclidean distances – that still remains NP-hard. The most remarkable of those that do are Arora’s
polynomial time (and even “nearly linear time”) approximation schemes (PTAS) [2,3], the good performance of
which is, however, only asymptotic. Other methods for Euclidean instances specifically include various heuristics
optimised for speed and based on clustering or partitioning of the plane, or spacefilling curves [
14
].
Here, we make another stab at formulating and finding (local) solutions to the Euclidean TSP. Our approach
consists of first reformulating the problem as a continuous diff-convex problem. Instead of attempting to find the
optimal path, we attempt to find points that construct the path, constrained to equal one of the input vertices.
We then relax this problem, converting the constraint into a mere penalty. Dependent on the formulation
of the constraint, the relaxed problem is found to be equivalent to certain clustering problems (including the
multisource Weber problem or “K-spatial medians”) perturbed with the path length penalty. (Perhaps not so
coincidentally, Arora’s methods can also be extended to approximate the K-spatial medians [
3,4
].)
As a continuation of the work in [
22,23
], in this paper we restrict ourselves to locally solving these penalised
reformulations, by applying the so-called “perturbed Weiszfeld method” applicable to finding “semi-critical”
points of a sum of Euclidean distances from fixed points, perturbed by a concave function. Although applicable
to the multisource Weber problem (providing a sort of dual of the K-means -style algorithm), it is unfortunately
not applicable to the problem perturbed with the path length penalty. The algorithm is, however, applicable
to another clustering formulation presented in [23], perturbed with the path length penalty. It is this latter
reformulation we will use in our numerical experiments.
An (approximate) solution of such a continuous reformulation of the Euclidean TSP is not in practice – and
not in theory either for big penalty parameters – a permutation of the original vertices. Therefore, along the
course of studying these reformulations, we derive a heuristic that we use to “associate” the points of a solution
with the original vertices. We also develop some other heuristics to reduce problem sizes, based on this heuristic
and the clustering principle.
As for the applicability of our algorithms, we do not have any theoretical proofs of efficiency aside from
partial convergence to “semi-critical points” (often local minima), and each step of the basic algorithm being
O(n2) (consisting of n parallel Weiszfeld steps). On the experimental side, our method does seem to provide
rather good results in quite few iterations for small problems. For bigger problems the performance however
degrades considerably – there are, after all, many more local solutions then. A bigger penalty parameter value
might help, but the algorithm we apply has a limit on its magnitude. Clustering heuristics that we develop,
however, somewhat remedy the situation. Nevertheless, our numerical results are not remarkable compared to
what is achievable with other (non-Euclidean) algorithms [
14
].
The primary contributions of this work are thus the reformulations that appear new, and perhaps with other
methods applied to them, could provide better numerical results. The basic method based on the Weiszfeld
algorithm is also new. Our clustering heuristics are related to the classic Karp clustering heuristic, Bentley’s
Fast Recursive Partitioning scheme [
8
], and Litke’s clustering heuristic [
15
]. The first two of these use a
“hardcoded” partitioning approach until the clusters are small enough, after which the sub-problems in the cluster are
solved either approximately o (...truncated)