using large-scale computation to estimate the beardwood-halton-hammersley tsp constant
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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). The Traveling Salesman Problem - PowerPoint PPT PresentationTRANSCRIPT
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)
Using large-scale computation to estimate the BHH TSP constantXLII SBPO, September 2, 2010
Outline
• The Traveling Salesman Problem
• The Beardwood-Halton-Hammersley theorem
• Past estimates of the BHH constant
• Our estimate
• Exploration of what affects convergence
Using large-scale computation to estimate the BHH TSP constantXLII SBPO, September 2, 2010
The Traveling Salesman Problem
Using large-scale computation to estimate the BHH TSP constantXLII SBPO, September 2, 2010
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.
Using large-scale computation to estimate the BHH TSP constantXLII SBPO, September 2, 2010
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?
Using large-scale computation to estimate the BHH TSP constantXLII SBPO, September 2, 2010
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.
Using large-scale computation to estimate the BHH TSP constantXLII SBPO, September 2, 2010
Optimal tour lengths
Using large-scale computation to estimate the BHH TSP constantXLII SBPO, September 2, 2010
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”
Using large-scale computation to estimate the BHH TSP constantXLII SBPO, September 2, 2010
Toroidal instance
A
A
B
B
Euclidean instance
Using large-scale computation to estimate the BHH TSP constantXLII SBPO, September 2, 2010
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
Using large-scale computation to estimate the BHH TSP constantXLII SBPO, September 2, 2010
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
Using large-scale computation to estimate the BHH TSP constantXLII SBPO, September 2, 2010
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
Using large-scale computation to estimate the BHH TSP constantXLII SBPO, September 2, 2010
“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
Using large-scale computation to estimate the BHH TSP constantXLII SBPO, September 2, 2010
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
Using large-scale computation to estimate the BHH TSP constantXLII SBPO, September 2, 2010
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
Using large-scale computation to estimate the BHH TSP constantXLII SBPO, September 2, 2010
Running times (in seconds) for 1,000,000 Concorde runs on random 1000-city “Toroidal” Euclidean instances
Range: 2.6 seconds to 6 hours
Using large-scale computation to estimate the BHH TSP constantXLII SBPO, September 2, 2010
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
Using large-scale computation to estimate the BHH TSP constantXLII SBPO, September 2, 2010
Euclidean vs Toroidal
Using large-scale computation to estimate the BHH TSP constantXLII SBPO, September 2, 2010
Toroidal (zoomed in)
Using large-scale computation to estimate the BHH TSP constantXLII SBPO, September 2, 2010
Residuals
Using large-scale computation to estimate the BHH TSP constantXLII SBPO, September 2, 2010
Provisional result
β ≈ 0.712403 ± 0.000007
BUT• Guessing functional form for fit
• ∞ is extreme extrapolation
• Strange behavior for small n
Using large-scale computation to estimate the BHH TSP constantXLII SBPO, September 2, 2010
Strange behavior for small n
Using large-scale computation to estimate the BHH TSP constantXLII SBPO, September 2, 2010
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?
Using large-scale computation to estimate the BHH TSP constantXLII SBPO, September 2, 2010
TSP – degree 2 = spanning tree
Using large-scale computation to estimate the BHH TSP constantXLII SBPO, September 2, 2010
TSP – connected = 2-factor(vertex cover by cycles)
Using large-scale computation to estimate the BHH TSP constantXLII SBPO, September 2, 2010
TSP – integer = subtour bound
Using large-scale computation to estimate the BHH TSP constantXLII SBPO, September 2, 2010
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?
Using large-scale computation to estimate the BHH TSP constantXLII SBPO, September 2, 2010
Euclidean square
• Not flatcorners and edges
• No translational symmetry
• No rotational symmetry
Using large-scale computation to estimate the BHH TSP constantXLII SBPO, September 2, 2010
Toroidal square
• Mostly flatup to r=0.5, πr2≈0.78
• Translational symmetry
• no Rotational symmetry
A
A
B
B
Using large-scale computation to estimate the BHH TSP constantXLII SBPO, September 2, 2010
Projective square
• Not flatcorners, but flatterthan euclidean
• No Translational symmetry
• No Rotational symmetry
A
A
B
B
Using large-scale computation to estimate the BHH TSP constantXLII SBPO, September 2, 2010
Klein square
• Mostly flatup to r=0.5
• no Translational symmetry
• no Rotational symmetry
A
A
B
B
Using large-scale computation to estimate the BHH TSP constantXLII SBPO, September 2, 2010
Toroidal hexagon
• Not flat, but flatterup to r≈0.537, πr2≈0.91
• Translational symmetry
• No Rotational symmetry
A
A
B
B
C
C
Using large-scale computation to estimate the BHH TSP constantXLII SBPO, September 2, 2010
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
Using large-scale computation to estimate the BHH TSP constantXLII SBPO, September 2, 2010
Sphere S2
• 2-d surface of 3-d sphere
• Great-circle (geodesic) distance
• Not flat, except in the limit
• Translational symmetry
• Rotational symmetry
Using large-scale computation to estimate the BHH TSP constantXLII SBPO, September 2, 2010
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
• Translational symmetry
• Rotational symmetry
Using large-scale computation to estimate the BHH TSP constantXLII SBPO, September 2, 2010
Topology and convergencecircles & spheres
Using large-scale computation to estimate the BHH TSP constantXLII SBPO, September 2, 2010
Topology and convergencemostly flat
Using large-scale computation to estimate the BHH TSP constantXLII SBPO, September 2, 2010
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
Using large-scale computation to estimate the BHH TSP constantXLII SBPO, September 2, 2010
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?
Using large-scale computation to estimate the BHH TSP constantXLII SBPO, September 2, 2010
Thank you