Using large-scale
computation to estimate the
Beardwood-HaltonHammersley 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 constant
XLII 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 constant
XLII SBPO, September 2, 2010
The Traveling Salesman Problem
Using large-scale computation to estimate the BHH TSP constant
XLII 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 constant
XLII SBPO, September 2, 2010
Beardwood, Halton, and Hammersley
The expected optimal tour length for an n-city instance
approaches βn 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 constant
XLII SBPO, September 2, 2010
The BHH constant
Early 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 constant
XLII SBPO, September 2, 2010
Optimal tour lengths
Using large-scale computation to estimate the BHH TSP constant
XLII 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 constant
XLII SBPO, September 2, 2010
Toroidal
Euclideaninstance
instance
B
B
A
A
Using large-scale computation to estimate the BHH TSP constant
XLII 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 constant
XLII SBPO, September 2, 2010
Toroidal estimates
Percus & 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 constant
XLII 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 constant
XLII SBPO, September 2, 2010
“Toroidal” estimate
Jacobsen, 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 constant
XLII 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 constant
XLII 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 constant
XLII 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 constant
XLII SBPO, September 2, 2010
Toroidal data points
Number of Cities
Number of
Instances
OPT
SUBTOUR
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
1,000
X
100
X
n = 100,000
n = 1,000,000
Using large-scale computation to estimate the BHH TSP constant
XLII SBPO, September 2, 2010
Euclidean vs Toroidal
Using large-scale computation to estimate the BHH TSP constant
XLII SBPO, September 2, 2010
Toroidal (zoomed in)
Using large-scale computation to estimate the BHH TSP constant
XLII SBPO, September 2, 2010
Residuals
Using large-scale computation to estimate the BHH TSP constant
XLII 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 constant
XLII SBPO, September 2, 2010
Strange behavior for small n
Using large-scale computation to estimate the BHH TSP constant
XLII 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 constant
XLII SBPO, September 2, 2010
TSP – degree 2 = spanning tree
Using large-scale computation to estimate the BHH TSP constant
XLII SBPO, September 2, 2010
TSP – connected = 2-factor
(vertex cover by cycles)
Using large-scale computation to estimate the BHH TSP constant
XLII SBPO, September 2, 2010
TSP – integer = subtour bound
Using large-scale computation to estimate the BHH TSP constant
XLII 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 constant
XLII SBPO, September 2, 2010
Euclidean square
• Not flat
corners and edges
• No translational symmetry
• No rotational symmetry
Using large-scale computation to estimate the BHH TSP constant
XLII SBPO, September 2, 2010
Toroidal square
A
• Mostly flat
up to r=0.5, πr2≈0.78
• Translational symmetry
B
B
• no Rotational symmetry
A
Using large-scale computation to estimate the BHH TSP constant
XLII SBPO, September 2, 2010
Projective square
A
B
• Not flat
corners, but flatter
than euclidean
• No Rotational symmetry
B
• No Translational symmetry
A
Using large-scale computation to estimate the BHH TSP constant
XLII SBPO, September 2, 2010
Klein square
A
• Mostly flat
up to r=0.5
• no Translational symmetry
B
B
• no Rotational symmetry
A
Using large-scale computation to estimate the BHH TSP constant
XLII SBPO, September 2, 2010
Toroidal hexagon
A
• Not flat, but flatter
up to r≈0.537, πr2≈0.91
• Translational symmetry
• No Rotational symmetry
A
Using large-scale computation to estimate the BHH TSP constant
XLII SBPO, September 2, 2010
Projective disc
A
• Not flat
• No translational symmetry
• Rotational symmetry
• Distance function hard
reflection in circular mirror
– Al-hazen’s problem
– reduces to solving quartic equation
–
A
Using large-scale computation to estimate the BHH TSP constant
XLII 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 constant
XLII 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 constant
XLII SBPO, September 2, 2010
Topology and convergence
circles & spheres
Using large-scale computation to estimate the BHH TSP constant
XLII SBPO, September 2, 2010
Topology and convergence
mostly flat
Using large-scale computation to estimate the BHH TSP constant
XLII 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 constant
XLII 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 constant
XLII SBPO, September 2, 2010
Thank you
Using large-scale computation to estimate the BHH TSP constant
XLII SBPO, September 2, 2010