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

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

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

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  1. 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)

  2. The Traveling Salesman Problem The Beardwood-Halton-Hammersley theorem Past estimates of the BHH constant Our estimate Exploration of what affects convergence Outline

  3. The Traveling Salesman Problem

  4. 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. Random Euclidean Instances

  5. 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? Beardwood, Halton, and Hammersley

  6. 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. The BHH constantEarly estimates

  7. Optimal tour lengths

  8. 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” Estimates fitting β + a/√n

  9. B A A B Euclidean instance Toroidal instance

  10. 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 advantages

  11. 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 Percus & Martin (1996)

  12. 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)/βHKbased 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 estimates Johnson, McGeogh, Rothberg (1996)

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

  14. 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 β Estimate summary

  15. 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 What’s new?

  16. Running times (in seconds) for 1,000,000 Concorde runs on random 1000-city “Toroidal” Euclidean instances Range: 2.6 seconds to 6 hours

  17. Toroidal data points

  18. Euclidean vs Toroidal

  19. Toroidal (zoomed in)

  20. Residuals

  21. β ≈ 0.712403 ± 0.000007 BUT Guessing functional form for fit ∞ is extreme extrapolation Strange behavior for small n Provisional result

  22. Strange behavior for small n

  23. 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? What affects convergence?

  24. TSP – degree 2 = spanning tree

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

  26. TSP – integer = subtour bound

  27. 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? What affects convergence?

  28. Not flat corners and edges No translational symmetry No rotational symmetry Euclidean square

  29. Mostly flatup to r=0.5, πr2≈0.78 Translational symmetry no Rotational symmetry B A A B Toroidal square

  30. Not flat corners, but flatter than euclidean No Translational symmetry No Rotational symmetry A B B A Projective square

  31. Mostly flat up to r=0.5 no Translational symmetry no Rotational symmetry B A B A Klein square

  32. Not flat, but flatterup to r≈0.537,πr2≈0.91 Translational symmetry No Rotational symmetry B B A A C C Toroidal hexagon

  33. 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 Projective disc

  34. 2-d surface of 3-d sphere Great-circle (geodesic) distance Not flat, except in the limit Translational symmetry Rotational symmetry Sphere S2

  35. 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 Projective Sphere

  36. Topology and convergence circles & spheres

  37. Topology and convergence mostly flat

  38. 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 Conclusions

  39. 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? Open questions

  40. Thank you

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