# New Bounds for the Traveling Salesman Constant

@article{Steinerberger2015NewBF, title={New Bounds for the Traveling Salesman Constant}, author={Stefan Steinerberger}, journal={Advances in Applied Probability}, year={2015}, volume={47}, pages={27 - 36} }

Let X 1, X 2, …, X n be independent and uniformly distributed random variables in the unit square [0, 1]2, and let L(X 1, …, X n ) be the length of the shortest traveling salesman path through these points. In 1959, Beardwood, Halton and Hammersley proved the existence of a universal constant β such that lim n→∞ n −1/2 L(X 1, …, X n ) = β almost surely. The best bounds for β are still those originally established by Beardwood, Halton and Hammersley, namely 0.625 ≤ β ≤ 0.922. We slightly improve…

## 32 Citations

An improved lower bound for the Traveling Salesman constant

- Mathematics, Computer ScienceOper. Res. Lett.
- 2020

The lower bound for the constant $\beta$ is improved to 0.6277 by an approach proposed in (Steinerberger 2015), which is based on the analysis in (Beardwood 1959).

Separating subadditive euclidean functionals

- Mathematics, Computer ScienceSTOC
- 2016

It is proved that the minimum length TSP on random points in Euclidean space is indeed asymptotically distinct from these and other natural lower bounds, and this separation implies that branch-and-bound algorithms based on these natural lower limits must take nearly exponential time to solve the TSP to optimality, even in average case.

Probabilistic Analysis of Euclidean Capacitated Vehicle Routing

- Computer ScienceISAAC
- 2021

It is shown that when k = √ n, the ITP algorithm is at best a (1 + c0)-approximation for some positive constant c0, which is a new lower bound on the optimal cost for the metric capacitated vehicle routing problem, which may be of independent interest.

Scalefree hardness of average-case Euclidean TSP approximation

- Mathematics, Computer ScienceArXiv
- 2016

It is shown that when using a heuristic from this class, a natural class of branch-and-bound algorithms takes exponential time to find an optimal tour (again, even on a random point-set), regardless of the particular branching strategy or lower-bound algorithm used.

Bounds for the traveling salesman paths of two-dimensional modular lattices

- Mathematics, Computer ScienceJ. Comb. Optim.
- 2017

Tight upper and lower bounds are presented for the traveling salesman path through the points of two-dimensional modular lattices based on earlier work on shortest vectors in lattices as well as on the strong convergence of Jacobi–Perron type algorithms.

Comb inequalities for typical Euclidean TSP instances

- Mathematics, Computer ScienceArXiv
- 2020

We prove that even in average case, the Euclidean Traveling Salesman Problem exhibits an integrality gap of $(1+\epsilon)$ for $\epsilon>0$ when the Held-Karp Linear Programming relaxation is…

Separating subadditive euclidean functionals

- Mathematics, Computer ScienceRandom Struct. Algorithms
- 2017

It is proved that a certain natural class of simple algorithms cannot solve the random Euclidean TSP efficiently, and asymptotically separate the TSP from its linear programming relaxation in this setting.

PROBABILITY DISTRIBUTION OF THE LENGTH OF THE SHORTEST TOUR BETWEEN A FEW RANDOM POINTS : A SIMULATION STUDY

- 2018

Inspired by an application in the field of on-demand public transportation, we perform a Monte Carlo simulation study on the probability distribution of the length of Traveling-Salesman-Problem (TSP)…

PROBABILITY DISTRIBUTION OF THE LENGTH OF THE SHORTEST TOUR BETWEEN A FEW RANDOM POINTS: A SIMULATION STUDY

- Mathematics, Computer Science2018 Winter Simulation Conference (WSC)
- 2018

It is shown that, under certain assumptions on the shape of the region and the probability distribution of locations, the length of the TSP tour is well-approximated by a normal distribution, even for as few as five locations.

Randomized Near Neighbor Graphs, Giant Components, and Applications in Data Science

- Mathematics, Computer ScienceJ. Appl. Probab.
- 2020

It is proved that it suffices to connect every point to c d,1 log log n points chosen randomly among its cd,2 log n-nearest neighbors to ensure a giant component of size n - o(n) with high probability.

## References

SHOWING 1-10 OF 27 REFERENCES

The shortest path through many points

- Mathematics
- 1959

We prove that the length of the shortest closed path through n points in a bounded plane region of area v is ‘almost always’ asymptotically proportional to √( nv ) for large n ; and we extend this…

Asymptotic experimental analysis for the Held-Karp traveling salesman bound

- Mathematics, Computer ScienceSODA '96
- 1996

Empirical evidence is provided in support of using theHK bound as a stand-in for the optimal tour length when evaluating the quality of near-optimal tours, and data indicates that the HK bound can provide substantial ‘‘variance reduction’’ in experimental studies involving randomly generated instances.

The Minimum Spanning Tree Constant in Geometrical Probability and Under the Independent Model: A Unified Approach

- Mathematics
- 1992

Given n uniformly and independently points in the d dimensional cube of unit volume, it is well established that the length of the minimum spanning tree on these n points is asymptotic to…

The shortest path and the shortest road through n points

- Mathematics
- 1955

Consider a set of n points lying in a square of side 1. Verblunsky has shown that, if n is sufficiently large, there is some path through all n points whose length does not exceed (2·8 n ) 1/2 +2. L.…

ON THE NUMBER OF LEAVES OF A EUCLIDEAN MINIMAL SPANNING TREE

- Mathematics
- 1987

Let V k,n be the number of vertices of degree k in the Euclidean minimal spanning tree of X i , , where the X i are independent, absolutely continuous random variables with values in R d . It is…

Estimating the Held-Karp lower bound for the geometric TSP

- Mathematics
- 1997

The Held-Karp lower bound (HK) provides a very good problem-specific estimate of optimal tour length for the travelling salesman problem (TSP). This measure, which is relatively quick and easy to…

Subadditive Euclidean Functionals and Nonlinear Growth in Geometric Probability

- Mathematics
- 1981

A limit theorem is established for a class of random processes (called here subadditive Euclidean functionals) which arise in problems of geometric probability. Particular examples include the length…

Probability theory and combinatorial optimization

- Mathematics
- 1987

Preface 1. First View of Problems and Methods. A first example. Long common subsequences Subadditivity and expected values Azuma's inequality and a first application A second example. The…

The traveling salesman problem and its variations

- Mathematics
- 2007

Preface. Contributing Authors.- 1. The Traveling Salesman Problem: Applications, Formulations and Variations.- 2. Polyhedral Theory and Branch-and-Cut Algorithms for the Symmetric TSP.- 3. Polyhedral…