The.Algorithm.Design.Manual.Springer-Verlag.1998
The.Algorithm.Design.Manual.Springer-Verlag.1998 The.Algorithm.Design.Manual.Springer-Verlag.1998
Traveling Salesman Problem Pascal implementations of branch-and-bound search and the insertion and Kerighan-Lin heuristics (for 2- opt and 3-opt) appear in [SDK83]. For details, see Section . Algorithm 608 [Wes83] of the Collected Algorithms of the ACM is a Fortran implementation of a heuristic for the quadratic assignment problem, a more general problem that includes the traveling salesman as a special case. Algorithm 750 [CDT95] is a Fortran code for the exact solution of asymmetric TSP instances. See Section for details. XTango (see Section ) includes animations of both the minimum spanning tree heuristic and a genetic algorithm for TSP. The latter converges sufficiently slowly to kill one's interest in genetic algorithms. Combinatorica [Ski90] provides (slow) Mathematica implementations of exact and approximate TSP solutions. See Section . Notes: The definitive reference on the traveling salesman problem is the book by Lawler et. al. [LLKS85]. Experimental results on heuristic methods for solving large TSPs include [Ben92a, GBDS80, Rei94]. Typically, it is possible to get within a few percent of optimal with such methods. TSPLIB [Rei91] provides the standard collection of hard instances of TSPs that arise in practice. The Christofides heuristic is an improvement of the minimum-spanning tree heuristic and guarantees a tour whose cost is at most 3/2 times optimal on Euclidean graphs. It runs in , where the bottleneck is the time it takes to find a minimum-weight perfect matching (see Section ). Good expositions of the Christofides heuristic [Chr76] include [Man89, PS85]. Expositions of the minimum spanning tree heuristic [RSL77] include [CLR90, O'R94, PS85]. Polynomial-time approximation schemes for Euclidean TSP have been recently developed by Arora [Aro96] and Mitchell [Mit96], which offer factor approximations in polynomial time for any . They are of great theoretical interest, although any practical consequences remain to be determined. The history of progress on optimal TSP solutions is somewhat inspiring. In 1954, Dantzig, Fulkerson, and Johnson solved a symmetric TSP instance of 42 United States cities [DFJ54]. In 1980, Padberg and Hong solved an instance on 318 vertices [PH80]. Applegate et. al. [ABCC95] have recently solved problems that are twenty times larger than this. Some of this increase is due to improved hardware, but most is due to better algorithms. The rate of growth demonstrates that exact solutions to NP-complete problems can be obtained for large instances if the stakes are high enough. Unfortunately, they seldom are. Good expositions on branch-and-bound methods include [PS82, SDK83]. Good expositions of the Kernighan-Lin heuristic [LK73, Lin65] include [MS91, PS82, SDK83]. file:///E|/BOOK/BOOK4/NODE175.HTM (4 of 5) [19/1/2003 1:31:21]
Traveling Salesman Problem Size is not the only criterion for hardness. One can easily construct an enormous graph consisting of one cheap cycle, for which it would be easy to find the optimal solution. For sets of points in convex position in the plane, the minimum TSP tour is described by its convex hull (see Section ), which can be computed in time. Other easy special cases are known. Related Problems: Hamiltonian cycle (see page ), minimum spanning tree (see page ), convex hull (see page ). Next: Hamiltonian Cycle Up: Graph Problems: Hard Problems Previous: Vertex Cover Algorithms Mon Jun 2 23:33:50 EDT 1997 file:///E|/BOOK/BOOK4/NODE175.HTM (5 of 5) [19/1/2003 1:31:21]
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Traveling Salesman Problem<br />
Size is not the only criterion for hardness. One can easily construct an enormous graph consisting of one<br />
cheap cycle, for which it would be easy to find the optimal solution. For sets of points in convex position<br />
in the plane, the minimum TSP tour is described by its convex hull (see Section ), which can be<br />
computed in time. Other easy special cases are known.<br />
Related Problems: Hamiltonian cycle (see page ), minimum spanning tree (see page ), convex<br />
hull (see page ).<br />
Next: Hamiltonian Cycle Up: Graph Problems: Hard Problems Previous: Vertex Cover<br />
<strong>Algorithm</strong>s<br />
Mon Jun 2 23:33:50 EDT 1997<br />
file:///E|/BOOK/BOOK4/NODE175.HTM (5 of 5) [19/1/2003 1:31:21]
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