Separating subadditive euclidean functionals

Alan Frieze,Wesley Pegden

doi:10.1145/2897518.2897571

Abstract

The classical Beardwood-Halton-Hammersly theorem (1959) asserts the existence of an asymptotic formula of the form constant times square root n for the minimum length of a Traveling Salesperson Tour through 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 the unit square 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.

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) [13]
We let M1 and M2 denote a pair of matchings derived from the minimum length TSP

Summary

Introduction

Halton, and Hammersley [3] studied the length of a Traveling Salesperson Tour through random points in Euclidean space. Considering Euclidean functionals MSTk(X) (with corresponding constants βMd STk ) defined as the minimum length of a spanning tree of X whose vertices all have degree. There are only finitely many constants βMd STk for each d; while we can draw trees with very large degrees, large degrees (relative to d) are not useful for minimum spanning trees in Euclidean space In contrast to this scenario, let us recall that a 2-factor is a disjoint set of cycles covering a given set of points. This class of algorithms includes many naturally occurring variants, and we will prove: Theorem 1.8. This gives a rigorous explanation for the observation (see [12], for example) that branch-and-bound heuristics using the Assignment Problem as a bounding estimate (even weaker than the 2-factor) perform poorly on random Euclidean instances

Subadditive Euclidean Functionals

Separating asymptotic constants

The 2-factor limit

Branch and Bound Algorithms

Final Remarks