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A Hard Problem: The TSP

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Graphs, Networks and Algorithms

Part of the book series: Algorithms and Computation in Mathematics ((AACIM,volume 5))

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Abstract

In this book, we have concentrated on those optimization problems which allow efficient (that is, polynomial time) algorithms. In contrast, the final chapter deals with an archetypical NP-complete problem: the travelling salesman problem already introduced in Chap. 1. It is one of the most famous and important problems in all of combinatorial optimization—with manyfold applications in such diverse areas as logistics, genetics, telecommunications, and neuroscience—and has been the subject of extensive study for about 60 years. We saw in Chap. 2 that no efficient algorithms are known for NP-complete problems, and that it is actually quite likely that no such algorithms can exist. Now we address the question of how such hard problems—which regularly occur in practical applications—might be approached: one uses, for instance, approximation techniques, heuristics, relaxations, post-optimization, local search, and complete enumeration. We shall explain these methods only for the TSP, but they are typical for dealing with hard problems in general. We will also brie y explain the idea of a further extremely important approach—via polyhedra—to solving hard problems and present a list of notable large scale TSPs which were solved to optimality.

Which way are you goin’…

Jim Croce

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Notes

  1. 1.

    Our discussion refers to the TSP, but applies to minimization problems in general. Of course, with appropriate adjustments, it can also be transferred to maximization problems.

  2. 2.

    In practice, this is done by using a sufficiently large number M instead of ∞: for instance, M=max{w ij :i,j=1,…,n}.

  3. 3.

    In the literature, it is quite common to assume s=1. Moreover, the term 1-tree is often used for the general concept (no matter which special vertex is selected), even though this is rather misleading.

  4. 4.

    We might solve MsT for each choice of s to obtain the best possible bound, but this is probably not worth the extra effort provided that we select s judiciously.

  5. 5.

    The polytope P is the convex hull of the incidence vectors of tours: its vertices are 0-1-vectors. Leaving out the restriction x ij ∈{0,1}, the inequalities in (15.1) and (15.2) define a polytope P′ containing P, which will (in general) have additional rational vertices. Thus all vertices of P′ which are not 0-1-vectors have to be cut off by further appropriate inequalities.

  6. 6.

    Note that this is the one point in the proof where we make use of the triangle inequality.

  7. 7.

    Of course we do not know such an optimal tour explicitly, but that does not matter for our argument.

  8. 8.

    Some other important problems are even more difficult to handle than the ΔTSP. For example, the existence of a polynomial ε-approximative algorithm for determining a maximal clique (for any particular choice of ε>0) already implies P=NP; see [AroSa02]. For even stronger results in this direction, we refer to [Zuc96]. All these results use an interesting concept from theoretical computer science: so-called transparent proofs; see, for example, [BabFL91] and [BabFLS91].

  9. 9.

    Obviously, k-opt needs O(n k) steps for each iteration of the while-loop; nothing can be said about the number of iterations required.

  10. 10.

    Of course, our previous experiences suggest that this idea will not be very helpful for the TSP; indeed, the present section will provide more bad news on the TSP in more than one respect. To use a phrase taken from [LawLRS85, p. 76]: The outlook continues to be bleak.

  11. 11.

    It can be shown that not even the huge neighborhood N n−3 of Sect. 15.6 is exact.

  12. 12.

    This fact is not all that surprising, as HaBe is the shortest edge incident with Be, while all the other edges are considerably longer.

  13. 13.

    Of course, it will usually be necessary to abort this process at some point: there is no guarantee at all that it has to terminate.

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    Dieter Jungnickel

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Jungnickel, D. (2013). A Hard Problem: The TSP. In: Graphs, Networks and Algorithms. Algorithms and Computation in Mathematics, vol 5. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32278-5_15

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