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on hard instances of the minimum weight triangulation
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On Hard Instances of the Minimum-Weight Triangulation Problem Sndor - - PowerPoint PPT Presentation

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On Hard Instances of the Minimum-Weight Triangulation Problem Sndor P. Fekete, Andreas Haas, Yannic Lieder, Eike Niehs, Michael Perk, Victoria Sack , and Christian Scheffer Department of Computer Science TU Braunschweig What is a

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On Hard Instances of the Minimum-Weight Triangulation Problem

SΓ‘ndor P. Fekete, Andreas Haas, Yannic Lieder, Eike Niehs, Michael Perk, Victoria Sack, and Christian Scheffer Department of Computer Science TU Braunschweig

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SLIDE 2

On Hard MWT Instances | Slide 2 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack, C. Scheffer

What is a Minimum-Weight Triangulation?

Wanted: A triangulation

  • f

that minimizes the length of its edges Given: A set

  • f

points in

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SLIDE 3

On Hard MWT Instances | Slide 3 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack, C. Scheffer

Solving an MWT-Instance Between 1979 and 2008

  • 1979/80: simple polygons (DP)

𝑃(π‘œ) [Gil79,Kli80]

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SLIDE 4

On Hard MWT Instances | Slide 4 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack, C. Scheffer

Solving an MWT-Instance Between 1979 and 2008

  • 1989: diamond property

[DJ89, DMS01]

  • 1979/80: simple polygons (DP)

𝑃(π‘œ) [Gil79,Kli80]

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SLIDE 5

On Hard MWT Instances | Slide 5 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack, C. Scheffer

Solving an MWT-Instance Between 1979 and 2008

  • 1989: diamond property

[DJ89, DMS01]

  • 1979/80: simple polygons (DP)

𝑃(π‘œ) [Gil79,Kli80]

  • 1997: LMT-skeleton

[DM96, DKM97]

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SLIDE 6

On Hard MWT Instances | Slide 6 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack, C. Scheffer

Solving an MWT-Instance Between 1979 and 2008

  • 1989: diamond property

[DJ89, DMS01]

  • 1979/80: simple polygons (DP)

𝑃(π‘œ) [Gil79,Kli80]

  • 1997: LMT-skeleton

[DM96, DKM97]

  • 2005/06: DP for polygons

with 𝑙 inner points

  • 𝑃(6π‘œ log π‘œ) [HO06]
  • 𝑃(π‘œ4𝑙) [GBL05a]
  • 𝑃(π‘œπ‘™! 𝑙) [GBL05b]
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SLIDE 7

On Hard MWT Instances | Slide 7 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack, C. Scheffer

MWT is NP-Hard

  • 2008: Proof by Mulzer and Rote [MR08]
  • Reduction from PLANAR 1-IN-3-SAT

all images from [MR08]

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SLIDE 8

On Hard MWT Instances | Slide 8 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack, C. Scheffer

The Gap

triangulate up to 30 Mio points in < 4min [Haa18] MWT is NP-hard

?

hard SAT-instances οƒ  millions of points οƒ  hardware as limiting factor

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SLIDE 9

On Hard MWT Instances | Slide 9 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack, C. Scheffer

The IP

β–³ β–³ β–³ β–³ β–³βˆˆ()

boundary component s.t.

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SLIDE 10

On Hard MWT Instances | Slide 10 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack, C. Scheffer

The IP

β–³ β–³ β–³ β–³ β–³βˆˆ()

boundary component s.t.

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SLIDE 11

On Hard MWT Instances | Slide 11 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack, C. Scheffer

The IP

β–³ β–³ β–³ β–³ β–³βˆˆ()

boundary component s.t.

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SLIDE 12

On Hard MWT Instances | Slide 12 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack, C. Scheffer

The IP

β–³ β–³ β–³ β–³ β–³βˆˆ()

boundary component s.t.

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SLIDE 13

On Hard MWT Instances | Slide 13 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack, C. Scheffer

The IP

β–³ β–³ β–³ β–³ β–³βˆˆ()

boundary component s.t.

β–³ β–³βˆˆ() β–³ β–³βˆˆ()

antennas

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SLIDE 14

On Hard MWT Instances | Slide 14 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack, C. Scheffer

The IP

β–³ β–³ β–³ β–³ β–³βˆˆ()

boundary component s.t.

β–³ β–³βˆˆ() β–³ β–³βˆˆ()

antennas

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SLIDE 15

On Hard MWT Instances | Slide 15 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack, C. Scheffer

The IP

β–³ β–³ β–³ β–³ β–³βˆˆ()

boundary component s.t.

β–³ β–³βˆˆ() β–³ β–³βˆˆ()

antennas

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SLIDE 16

On Hard MWT Instances | Slide 16 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack, C. Scheffer

The IP

β–³ β–³ β–³ β–³ β–³βˆˆ()

boundary component s.t.

β–³ β–³βˆˆ() β–³ β–³βˆˆ()

antennas

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SLIDE 17

On Hard MWT Instances | Slide 17 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack, C. Scheffer

The IP

β–³ β–³ β–³ β–³ β–³βˆˆ()

boundary component s.t.

β–³ β–³βˆˆ() β–³ β–³βˆˆ()

antennas

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SLIDE 18

On Hard MWT Instances | Slide 18 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack, C. Scheffer

The IP

β–³ β–³ β–³ β–³ β–³βˆˆ()

boundary component s.t.

β–³ β–³βˆˆ() β–³ β–³βˆˆ()

antennas

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SLIDE 19

On Hard MWT Instances | Slide 19 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack, C. Scheffer

The IP

β–³ β–³ β–³ β–³ β–³βˆˆ()

boundary component s.t.

β–³ β–³βˆˆ() β–³ β–³βˆˆ()

antennas

β–³ β–³ ∈() β–³ β–³βˆˆ

inner edges

β–³

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SLIDE 20

On Hard MWT Instances | Slide 20 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack, C. Scheffer

The IP

β–³ β–³ β–³ β–³ β–³βˆˆ()

boundary component s.t.

β–³ β–³βˆˆ() β–³ β–³βˆˆ()

antennas

β–³ β–³ ∈() β–³ β–³βˆˆ

inner edges

β–³

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SLIDE 21

On Hard MWT Instances | Slide 21 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack, C. Scheffer

The IP

β–³ β–³ β–³ β–³ β–³βˆˆ()

boundary component s.t.

β–³ β–³βˆˆ() β–³ β–³βˆˆ()

antennas

β–³ β–³ ∈() β–³ β–³βˆˆ

inner edges

β–³

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SLIDE 22

On Hard MWT Instances | Slide 22 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack, C. Scheffer

The IP

β–³ β–³ β–³ β–³ β–³βˆˆ()

boundary component s.t.

β–³ β–³βˆˆ() β–³ β–³βˆˆ()

antennas

β–³ β–³ ∈() β–³ β–³βˆˆ

inner edges

β–³

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SLIDE 23

On Hard MWT Instances | Slide 23 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack, C. Scheffer

The IP

β–³ β–³ β–³ β–³ β–³βˆˆ()

boundary component s.t.

β–³ β–³βˆˆ() β–³ β–³βˆˆ()

antennas

β–³ β–³ ∈() β–³ β–³βˆˆ

inner edges

β–³

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SLIDE 24

On Hard MWT Instances | Slide 24 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack, C. Scheffer

What Makes an MWT-Instance Practically Hard?

1. 1 complex face, else

  • 2. faces with many inner

points, else 𝑃 6π‘œ log π‘œ / 𝑃(π‘œ4𝑙) / 𝑃(π‘œπ‘™! 𝑙)

from [YY14] from [BKM96]

After Diamond-test and LMT-skeleton…

  • 3. LP-relaxation of

the IP must have fractional solution, else easy to solve

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On Hard MWT Instances | Slide 25 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack, C. Scheffer

Experimental Setup: Evolver

create initial instance generate/mutate

  • ffspring

evaluate fitness select next generation

construct small polygons in simple faces

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SLIDE 26

On Hard MWT Instances | Slide 26 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack, C. Scheffer

Experimental Setup: Evolver

construct small polygons in simple faces

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SLIDE 27

On Hard MWT Instances | Slide 27 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack, C. Scheffer

Experimental Setup: Pertubation

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On Hard MWT Instances | Slide 28 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack, C. Scheffer

Experimental Setup: Pertubation

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On Hard MWT Instances | Slide 29 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack, C. Scheffer

Experimental Setup: Pertubation

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On Hard MWT Instances | Slide 30 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack, C. Scheffer

Experimental Setup: Pertubation

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On Hard MWT Instances | Slide 31 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack, C. Scheffer

Experimental Setup: Pertubation

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On Hard MWT Instances | Slide 32 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack, C. Scheffer

Experimental Setup: Pertubation

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On Hard MWT Instances | Slide 33 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack, C. Scheffer

Experimental Setup: Pertubation

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SLIDE 34

On Hard MWT Instances | Slide 34 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack, C. Scheffer

Experimental Results: Runtime

  • created ~17,000 instances
  • fixed size of

306 points

No instance had a fractional LP solution!

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On Hard MWT Instances | Slide 35 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack, C. Scheffer

Experimental Results: Scaling

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On Hard MWT Instances | Slide 36 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack, C. Scheffer

Experimental Results: Stability

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On Hard MWT Instances | Slide 37 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack, C. Scheffer

Conclusion

What makes an MWT instance practically hard?

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On Hard MWT Instances | Slide 38 March 16, 2020 | S. P. Fekete, A. Haas, Y. Lieder, E. Niehs, M. Perk, V. Sack, C. Scheffer

References

[DJ89] Gautam Das and Deborah Joseph. Which triangulations approximate the complete graph? In Proc. International Symposium on Optimal Algorithms, pages 168–192, 1989. [DMS01] Robert L. Scot Drysdale, Scott A. McElfresh, and Jack Snoeyink. On exclusion regions for optimal triangulations. Discrete Applied Mathematics, 109(1-2):49–65, 2001. [DM96] Matthew Dickerson and Mark H. Montague. A (Usually?) Connected Subgraph of the Minimum Weight Triangulation. In Proc. Symposium on Computational Geometry (SoCG), pages 204–213, 1996. [DKM97] Matthew Dickerson, J. Mark Keil, and Mark H. Montague. A Large Subgraph of the Minimum Weight Triangulation. Discrete & Computational Geometry, 18(3):289–304, 1997. [Gil79]

  • P. D. Gilbert. New results in planar triangulations. Master’s thesis, University Illinois, 1979.

[Kli80] G.T. Klincsek. Minimal Triangulations of Polygonal Domains. Annals of Discrete Mathematics, 9:121–123, 1980. [HO06] Michael Hoffmann and Yoshio Okamoto. The minimum weight triangulation problem with few inner points. Computational Geometry, 34(3):149–158, 2006. [GBL05a] Magdalene Grantson, Christian Borgelt, and Christos Levcopoulos. A Fixed Parameter Algorithm for Minimum Weight Triangulation: Analysis and Experiments. Technical Report LU-CS-TR: 2005-234, Lund University, Sweden, 2005. [GBL05b] Magdalene Grantson, Christian Borgelt, and Christos Levcopoulos. Minimum Weight Triangulation by Cutting Out Triangles. In

  • Proc. International Symposium on Algorithms and Computationa (ISAAC), pages 984–994, 2005.

[MR08] Wolfgang Mulzer and GΓΌnter Rote. Minimum-weight triangulation is NP-hard. Journal of the ACM, 55(2):11, 2008. [Haa18] Andreas Haas. Solving large-scale minimum-weight triangulation instances to provable optimality. In Proc. Symposium on Computational Geometry (SoCG), pages 44:1–44:14, 2018. [BKM96] Patrice Belleville, Mark Keil, Michael McAllister, and Jack Snoeyink. On computing edges that are in all minimum-weight

  • triangulations. In Proc. Symposium on Computational Geometry (SoCG), pages V7–V8, 1996.

[YY14] Arman Yousefi and Neal E. Young. On a linear program for minimum-weight triangulation. SIAM Journal on Computing, 43(1):25–51, 2014.