Abstract
AbstractThe classical Beardwood‐Halton‐Hammersly theorem (1959) asserts the existence of an asymptotic formula of the form for the minimum length of a Traveling Salesperson Tour throuh n random points in the unit square, and in the decades since it was proved, the existence of such formulas has been shown for other such Euclidean functionals on random points in the unit square as well. Despite more than 50 years of attention, however, it remained unknown whether the minimum length TSP through n random points in was asymptotically distinct from its natural lower bounds, such as the minimum length spanning tree, the minimum length 2‐factor, or, as raised by Goemans and Bertsimas, from its linear programming relaxation. We prove that the TSP on random points in Euclidean space is indeed asymptotically distinct from these and other natural lower bounds, and show that this separation implies that branch‐and‐bound algorithms based on these natural lower bounds must take nearly exponential () time to solve the TSP to optimality, even in average case. This is the first average‐case superpolynomial lower bound for these branch‐and‐bound algorithms (a lower bound as strong as was not even been known in worst‐case analysis). © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 51, 375–403, 2017
Highlights
Halton, and Hammersley [3] studied the length of a Traveling Salesperson Tour through random points in Euclidean space
If x1, x2, . . . is a random sequence of points in [0, 1]d and Xn = {x1, . . . , xn}, their results imply that there is an absolute constant βTd SP such that the length TSP(Xn) of a minimum length tour through Xn satisfies βTd SPn d−1 d a.s. This result has many extensions; for example, we know that identical asymptotic formulas hold for the the cases of the minimum length of a spanning tree MST(Xn)[3], and the minimum length of a matching MM(Xn) [24]
MSTk d−1 d for each k, and we prove separation of these asymptotic formulas as follows: Theorem 1.2
Summary
Halton, and Hammersley [3] studied the length of a Traveling Salesperson Tour through random points in Euclidean space. W.h.p, the algorithm runs in time eΩ(n/ log n) This gives a rigorous explanation for the observation (see [22], for example) that branch-and-bound heuristics using the Assignment Problem as a bounding estimate (even weaker than the 2-factor) perform poorly on Euclidean instances, and indicates that the success of the Held-Karp bound in branch and bound algorithms will be limited for sufficiently large Euclidean point sets. We emphasize that this is the first average-case hardness result (stronger than worstcase hardness) for the Euclidean TSP of which we aware. Our results are all immediately valid in these more general settings, : as the constants βLd depend only on L and d or m (in particular, not on the distribution or, in the second case, the particular manifold), it is enough study the constants in the case of points which are uniformly distributed in the hypercube
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