The.Algorithm.Design.Manual.Springer-Verlag.1998

Traveling Salesman Problem
practice, it is usually better, typically 15% to 20% over optimal. Further, the time of the algorithm
is bounded by that of computing the minimum spanning tree, only in the case of points in
the plane (see Section ).
● Incremental insertion methods - A different class of heuristics inserts new points into a partial
tour one at a time (starting from a single vertex) until the tour is complete. The version of this
heuristic that seems to work best is furthest point insertion: of all remaining points, insert the
point v into partial tour T such that
The minimum ensures that we insert the vertex in the position that adds the smallest amount of
distance to the tour, while the maximum ensures that we pick the worst such vertex first. This
seems to work well because it first ``roughs out'' a partial tour before filling in details. Typically,
such tours are only 5% to 10% longer than optimal.
● k-optimal tours - Substantially more powerful are the Kernighan-Lin, or k-opt class of heuristics.
Starting from an arbitrary tour, the method applies local refinements to the tour in the hopes of
improving it. In particular, subsets of edges are deleted from the tour and the k remaining
subchains rewired in a different way to see if the resulting tour is an improvement. A tour is koptimal
when no subset of k edges can be deleted and rewired so as to reduce the cost of the tour.
Extensive experiments suggest that 3-optimal tours are usually within a few percent of the cost of
optimal tours. For k > 3, the computation time increases considerably faster than solution quality.
Two-opting a tour is a fast and effective way to improve any other heuristic. Simulated
annealing provides an alternate mechanism to employ edge flips to improve heuristic tours.
Implementations: The world-record-setting traveling salesman program is by Applegate, Bixby,
Chvatal, and Cook [ABCC95], which has solved instances as large as 7,397 vertices to optimality. At
this time, the program is not being distributed. However, the authors seem willing to use it to solve TSPs
sent to them. In their paper, they describe this work as neither theory nor practice, but sport - an almost
recreational endeavor designed principally to break records. It is a very impressive piece of work,
however.
The TSPLIB library of test instances for the traveling salesman problem is available from Netlib, and by
anonymous ftp from softlib.cs.rice.edu. See Section .
Tsp_solve is a C++ code by Chad Hurwitz and Robert Craig that provides both heuristic and optimal
solutions. Geometric problems of size up to 100 points are manageable. It is available from
http://www.cs.sunysb.edu/ algorith or by e-mailing Chad Hurrwitz at churritz@crash.cts.com. A
heuristic Euclidean TSP solver in C due to Lionnel Maugis is available from
http://www.cenaath.cena.dgac.fr/ maugis/tsp.shar.
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