Using large-scale computation to estimate the Beardwood-Halton-Hammersley TSP constant

Using large-scale computation to estimate the BHH TSP constantXLII SBPO, September 2, 2010

Using large-scale computation to estimate the

Beardwood-Halton-Hammersley TSP constant

David Applegate (AT&T Labs – Research)

William Cook (Georgia Tech)

David S. Johnson (AT&T Labs – Research)

Neil J. A. Sloane (AT&T Labs – Research)


Outline

• The Traveling Salesman Problem

• The Beardwood-Halton-Hammersley theorem

• Past estimates of the BHH constant

• Our estimate

• Exploration of what affects convergence


The Traveling Salesman Problem


Random Euclidean Instances

• Easy to generate, easy to draw, for arbitrary sizes.

• Performance of heuristics and optimization algorithms on these instances are reasonably well-correlated with that for real-world geometric instances.

• The canonical TSP test case.

• (technical note) To form integer objective and avoid problems comparing sums of square roots, we use 10^6 x 10^6 integer grid for points, and round edge lengths to integers.


Beardwood, Halton, and Hammersley

The expected optimal tour length for an n-city instance approaches βn for some constant β as n . [Beardwood, Halton, and Hammersley, 1959]

That is, E[OPT/√n] → β

Open question: what is the value of β?


The BHH constantEarly estimates

1959: Beardwood, Halton, and Hammersley: ≈0.75

hand solutions to a 202-city and a 400-city instance.

1977: Stein: ≈0.765

extensive simulations on 100-city instances.


Optimal tour lengths


Estimates fitting β + a/√n

• 1989: Ong & Huang estimate β ≤ .74, based on runs of 3-Opt

• 1994: Fiechter estimates β ≤ .73, based on runs of “parallel tabu search”

• 1994: Lee & Choi estimate β ≤ .721, based on runs of “multicanonical annealing”


Toroidal instance

A

A

B

B

Euclidean instance


Toroidal advantages

• No boundary effects

• Jaillet (1992): E[OPT/√n] → β also for toroidal instances (but result is still asymptotic)

• Lower variance of OPT for fixed n

• In practice, instances tend to be easier– more than makes up for more expensive distance computation


Toroidal estimatesPercus & Martin (1996)

• 250,000 samples, n = 12,13,14,15,16,17 (“Optimal” = best of 10 Lin-Kernighan runs)

• 10,000 samples with n = 30 (“Optimal” = best of 5 runs of 10-step-Chained-LK)

• 6,000 samples with n = 100 (“Optimal” = best of 20 runs of 10-step-Chained-LK)

• Fit to OPT/N = (β + a/n + b/n2)/(1+1/(8n))

• β .7120 ± .0002


Toroidal estimates Johnson, McGeogh, Rothberg (1996)

Observe that• the Held-Karp (subtour) bound also has an asymptotic

constant, i.e., HK/n βHK and is easier to compute than the optimal.

• (OPT-HK)/n has a substantially lower variance than either OPT or HK.

Estimate

• (β - βHK)/βHK based on instances with n = 100, 316, 1000 using Concorde for n ≤ 316 and Iterated Lin-Kernighan plus Concorde-based error estimates for n = 1000.

• βHK based on instances from n=100 to 316,228 using heuristics and Concorde-based error estimates

• β .7124 ± .0002


“Toroidal” estimateJacobsen, Read, and Saleur (2004)

• Instead of toroidal square, use a 1 x 100,000 cylinder – that is, only join the top and bottom of the unit square and stretch the width by a factor of 100,000.

• Set n = 100,000 W and generate 10 samples each for W = 1,2,3,4,5,6.

• Optimize by using dynamic programming, where only those cities within distance k of the frontier (~kw cities) can have degree 0 or 1, k = 4,5,6,7,8.

• Estimate true optimal as k .

• Estimate unit square constant as W .

• With n ≥ 100,000, assume no need for asymptotics in n

• β .7119


β Estimate summary

• 0.75 (1959) Beardwood, Halton, Hammersley

• 0.765 (1977) Stein

• 0.74 (1989) Ong & Huang

• 0.73 (1994) Fiechter

• 0.721 (1994) Lee & Choi

• 0.7120 ± 0.0002 (1996) Percus & Martin

• 0.7124 ± 0.0002 (1996) Johnson, McGeoch, Rothberg

• 0.7119 (2004) Jacobsen, Read, Saleur


What’s new?

• Cycles are much faster and cheaper

• Concorde is much better– TSP-solving code by Applegate, Bixby, Chvátal, Cook– Available at http://www.tsp.gatech.edu/concorde– Also computes subtour (Held-Karp) and other bounds– Hoos and Stϋtzle (2009)

• median running time for Euclidean instances ≈0.21 · 1.24194 n

• n=2000 ≈57 minutes• n=4500 ≈96 hours


Running times (in seconds) for 1,000,000 Concorde runs on random 1000-city “Toroidal” Euclidean instances

Range: 2.6 seconds to 6 hours


Toroidal data points

Number of CitiesNumber of Instances

OPTSUB-TOUR

n = 3, 4, …, 49, 50 1,000,000 X X

n = 60, 70, 80, 90, 100 1,000,000 X X

n = 200, 300, …, 1,000 1,000,000 X X

n = 110, 120, …, 2,000 10,000 X X

n = 2,000, 3,000, …, 10,000 1,000,000 X

n = 100,000 1,000 X

n = 1,000,000 100 X


Euclidean vs Toroidal


Toroidal (zoomed in)


Residuals


Provisional result

β ≈ 0.712403 ± 0.000007

BUT• Guessing functional form for fit

• ∞ is extreme extrapolation

• Strange behavior for small n


Strange behavior for small n


What affects convergence?

• Constraints: TSP is– Degree 2– Connected– Integer

• Topology– Translational symmetry (point-transitivity)

are all points equivalent– Rotational symmetry

are all directions equivalent– Flatness

Does the area of a ball of radius r = πr2?


TSP – degree 2 = spanning tree


TSP – connected = 2-factor(vertex cover by cycles)


TSP – integer = subtour bound


What affects convergence?

• Constraints: TSP is– Degree 2– Connected– Integer

• Topology– Translational symmetry (point-transitivity)

are all points equivalent– Rotational symmetry

are all directions equivalent– Flatness

Does the area of a ball of radius r = πr2?


Euclidean square

• Not flatcorners and edges

• No translational symmetry

• No rotational symmetry


Toroidal square

• Mostly flatup to r=0.5, πr2≈0.78

• Translational symmetry

• no Rotational symmetry

A

A

B

B


Projective square

• Not flatcorners, but flatterthan euclidean

• No Translational symmetry

• No Rotational symmetry

A

A

B

B


Klein square

• Mostly flatup to r=0.5

• no Translational symmetry

• no Rotational symmetry

A

A

B

B


Toroidal hexagon

• Not flat, but flatterup to r≈0.537, πr2≈0.91


• No Rotational symmetry

A

A

B

B

C

C


Projective disc

• Not flat

• No translational symmetry

• Rotational symmetry

• Distance function hard– reflection in circular mirror– Al-hazen’s problem– reduces to solving quartic equation

A

A


Sphere S2

• 2-d surface of 3-d sphere

• Great-circle (geodesic) distance

• Not flat, except in the limit




Projective Sphere

• Lines in 3-space through the origin

• equivalently, points on a hemisphere

• Distance between lines is angle between them

• Not flat, except in the limit




Topology and convergencecircles & spheres


Topology and convergencemostly flat


Conclusions

β ≈ 0.712403 ± 0.000007

• Constraints affect β

• Topology affects convergence– Flatness matters a lot– Translational and rotational symmetry only matter a little– Topology doesn’t account for the behavior for small n


Open questions

• What is the 2nd order term in convergence

• Is decrease towards limit provable?

• What explains peak around n=17?

• Can the link between flatness and E[OPT(n)] be made more precise?


Thank you

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Using large-scale computation to estimate the Beardwood-Halton-Hammersley TSP constant

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