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A Partition-Based Heuristic for Translational Box Covering Ben England and Karen Daniels Department of Computer Science University of Massachusetts Lowell supported in part by NSF and DARPA under grant DMS-0310589 Sensor coverage: Repair work: collection of pieces cover a hole

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a partition based heuristic for translational box covering

A Partition-Based Heuristic for Translational Box Covering

Ben England and Karen Daniels

Department of Computer Science

University of Massachusetts Lowell

supported in part by NSF and DARPA under grant DMS-0310589

motivation for 2d polygonal covering
Sensor coverage:

Repair work: collection of pieces cover a hole

Motivation for 2D Polygonal Covering

NP-hard problem

Supported under NSF/DARPA CARGO program

box covering
Box Covering
  • Goal: Translate a collection of boxes (orthotopes) Q = {Q1, Q2 , ... , QN} to cover another box P in 2d, 3d, …
  • Motivation: Boxes can form enclosures for general shapes.

2d views of 3d covering

1st published results in > 2d

Partial cover (red part uncovered)

Full cover

40 covering shapes

20 covering shapes

d = dimension

NP-hard problem

With Masters student B. England

Supported under NSF/DARPA CARGO program

selected prior covering work

covering

. . . .

. . .

. . .

  • Survey of non-algorithmic results [Tot04]
  • Thin coverings of the plane with congruent convex shapes
  • Translational covering of arbitrary polygonal shapes [Dan01,Dan03]
  • Translational B-spline covering [Nea06]
  • Volume condition for translational covering of a cube by a sequence of convex shapes (arbitrary dimension) [Gro85]
  • Volume condition for on-line algorithm for translational covering of a cube by a sequence of convex shapes (arbitrary dimension) [Las97]
Selected Prior Covering Work

covering

combinatorial covering

geometric covering

VERTEX-COVER, SET-COVER (including [Gri99]), EDGE-COVER, VLSI logic minimization, facility location

translational covering

P: finite point sets

P: shapes

Q: identical

Q: nonconvex

Q: convex

. . .

decomposition:

1D interval covered by annuli using approximation algorithm [Hoc87]

BOX-COVER [Fow81]

NP-complete

partition:

decomposition with covering

  • NP-hardness proofs for 4 polygon covering problems [Cul88]
  • Approximation algorithms for some orthogonal covering problems [Ber92]
  • Approximation algorithm to cover orthogonal polygon (with holes) with minimum number of rectangles [Kum03]
  • Clique-based Integer Programming (IP) model for covering orthogonal polygon with minimum number of rectangles [Hei05]
  • Polynomial-time results for restricted orthogonal polygon covering and horizontally convex polygons

Polynomial-time algorithms for triangulation [Cha91] and some tilings

box covering outline
Box Covering Outline
  • Set covering approach
  • Key volume expressions
  • Partition-based heuristic
  • Experimental highlights
  • Dimension-independent volume test
  • Computational considerations
    • Execution time dominated by 1-OPT
    • Alternatives to 1-OPT
    • 1-OPT preprocessing
    • Monotonicity across calls to Lagrangian heuristic
  • Conclusion and future work

Lagrangian heuristic comes from Lagrangian relaxation of IP model.

1-OPT heuristic swaps groups for cover shape that best improves objective function until no improvement.

set covering approach applied to box covering

g12 = {2,4}

g11 = {1,3}

g22 = {3,4}

g21 = {1,2}

g32 = {2}

g31 = {1}

g34 = {4}

g33 = {3}

C1 = {g11,g12}

C2 = {g21,g22}

C3 = {g31,g32,g33,g34}

Problem: choose just one part group gjk from the set of part groups Cj for each cover shape Qj such that every part of P is in one of the chosen part groups.

Solution: {g11,g21,g34}

Set Covering Approach Applied to Box Covering

parts

IP model, maximizing number of parts covered, treated with Lagrangian Heuristic + 1-OPT

Based on Daniels and Grinde, IIE Transactions, 1999

key volume expressions
Key Volume Expressions

quantized volume

effective volume

quantized effective volume

volume

d = a generic part of P

slide8

Heuristic:

Uniform refinement scheme, unlike general polygonal approach of [Dan03], which subdivides one triangle during each iteration of repeat loop.

d = dimension

j = total number of parts of P

d = a generic part of P

N = number of covering shapes

LGC_Cover( ) = modified Lagrangian Heuristic + 1-OPT

experimental highlights

Minkowski sum of two sets A and B is

Experimental Highlights

2D Validation Experiment:

  • 20 instances with square P and N = 2…6 rectangular covering shapes
  • OrthotopeCover( ) outperforms polygonal solver of Daniels, et al. [CCCG2003] by at least 2 orders of magnitude
    • Simpler geometric operations (no Minkowski sum)
    • Volume tests that do not generalize to arbitrary polygons

Example:

4 identical square covering shapes

Daniels, et al. [CCCG2003]:

Current paper:

167 triangular parts

4 square parts

94 triangle vertices

5 square vertices

888 groups

80 groups

875 seconds

0.2 seconds

450 MHz CPU Sun SPARC Ultra 60TM with 512 MB memory

cover shape volume= = 1.21

experimental highlights10
Experimental Highlights

Results in 3d and 4d:

d = dimension

N = number of covering shapes

3 GHz 64-bit Intel PentiumTM D CPU with 2 GB memory

csv = cover shape volume

j = total number of parts of P

maximum aspect ratio of a covering shape = 4

experimental highlights11

2d views of 3d covering using OpenGL

Experimental Highlights

More results in 3d:

1 GHz Intel PentiumTM 4 CPU with 1/2 GB memory

problem instance hardness characterization
Problem Instance “Hardness” Characterization

N = number of covering shapes

d = dimension

d = a generic part of P

j = total number of parts of P

Quantized Effective Volume Ratio

Dimension-independent Volume Margin

slide13

Effectiveness of Y

~10 instances for each parameter combination

tp = total points.

r = correlation coefficient between Y and % coverage

1 GHz Intel PentiumTM 4 CPU with 1/2 GB memory

slide14

d # jN #calls %savings

instances saved

47.2

3

17

8192

2.8

12-18

128-4096

2.3

12-16

58.8

3

13

test added

Heuristic:

Effectiveness of Y

# calls saved = average per instance

% savings = average relative % savings of LCG_Cover( ) calls

100% coverage was reached by original & revised heuristic in all these cases.

3 GHz 64-bit Intel PentiumTM D CPU with 2 GB memory

computational considerations
Computational Considerations
  • Execution Time
    • OrthotopeCover( ) dominated by LGC_Cover( )
    • LGC_Cover( ) dominated by deterministic 1-OPT
      • 1-OPT attempts to increase lower bound on Lagrangian dual
    • Unlike polygonal heuristic in which group maintenance dominates
  • Alternatives to 1-OPT
    • 2-OPT too expensive
    • Randomization:
      • Simulated annealing’s random swaps inferior to 1-OPT
      • Random group sampling weakens 1-OPT
  • 1-OPT Preprocessing
    • 1-OPT really behaves like a greedy global improvement strategy
    • 1-OPT preprocessing yields improvement in:
      • 75% of 2d instances
      • 87% of 3d instances
      • 64% of 4d instances

Test suite = subset of 30 of our randomly generated instances: 10 2d, 10 3d, 10 4d

3 GHz 64-bit Intel PentiumTM D CPU with 2 GB memory

1-OPT heuristic swaps groups for cover shape that best improves objective function until no improvement.

computational considerations16
Computational Considerations
  • Monotonicity across calls to LGC_Cover( ) Lagrangian heuristic
    • No theoretical guarantee that number of parts covered increases.
      • Number of parts doubles before each LGC_Cover( ) call.
      • LGC_Cover( ) is only a heuristic.
    • Success depends on N, d, thickness of cover, richness of group structure and strength of LGC_Cover( ).
      • Group structure is rich.
      • 1-OPT helps LGC_Cover( ) cover increasing number of parts.
    • Sample progression for 2d, N = 6, csv = 1.25:
      • 504 of 512 parts covered (98.4%)
      • 1014 of 1024 parts covered (99.%)
      • 2039 of 2048 parts covered (99.6%)
      • 4096 of 4096 parts covered (100%)

monotonically increasing

Test suite = subset of 30 of our randomly generated instances: 10 2d, 10 3d, 10 4d

3 GHz 64-bit Intel PentiumTM D CPU with 2 GB memory

1-OPT heuristic swaps groups for cover shape that best improves objective function until no improvement.

conclusion future work
Conclusion & Future Work
  • Partition-based, translational, box covering heuristic has dimension as an input.
  • Set covering approach uses uniform refinement.
  • First 3d, 4d heuristic results for our translational box covering problem
    • Found covers for some instances with as many as 50 covering shapes.
  • Box covering heuristic outperforms general, polygonal heuristic in 2d rectangular experiment.
  • Dimension-independent volume margin avoids many refinement steps.
  • Computational considerations
    • Execution time is dominated by deterministic 1-OPT improvement heuristic.
    • 1-OPT outperforms 2-OPT, simulated annealing and randomized 1-OPT.
    • 1-OPT preprocessing improves results.
    • Monotonicity across calls to Lagrangian heuristic occurs often in practice, although not theoretically guaranteed.
  • Future work:
    • Use boxes as enclosures for more general shapes to:
      • improve 2d general covering heuristic
      • treat 3d general covering
    • Allow rotations
references
References

Acknowledgement: Thanks to Michelle Daniels for comments.

for more information
For More Information
  • Email kdaniels@cs.uml.edu
  • Web sites:
  • Thanks for your attention!
  • Questions?

http://www.cs.uml.edu/~kdaniels/covering/covering.htm

http://www.cs.uml.edu/~bengland/cg/orthotope_cover/

19

backup slides

BACKUP SLIDES

(from CCCG 2003, etc.)

2d polygonal covering cccg 2001 cccg2003

Translational 2D Polygon Covering

P2

P2

P1

Q3

P1

Q2

Q2

Q1

Sample P and Q

Translated Q Covers P

Q1

Q3

2D Polygonal Covering [CCCG 2001,CCCG2003]

Supported under NSF/DARPA CARGO program

  • Input:
    • Covering polygons Q = {Q1, Q2 , ... , Qm}
    • Target polygons (or point-sets) P = {P1, P2 , ... , Pn}
  • Output:
    • Translations g = {g1, g2 , ... , gm} such that

With graduate students R. Inkulu, A. Mathur, C.Neacsu, & UNH professor R. Grinde

slide22
2D B-Spline Covering [CORS/INFORMS2004, UMass Lowell Student Research Symposium 2004, Computers Graphics Forum, 2006]

Supported under NSF/DARPA CARGO program

motivated by 3D CAD

NP-hard problem

With graduate student C. Neacsu

covering web site http www cs uml edu kdaniels covering covering htm
Covering Web Sitehttp://www.cs.uml.edu/~kdaniels/covering/covering.htm

With graduate student C. Neacsu and undergraduate A. Hussin

combinatorial covering procedure lagrangian cover ip model
Combinatorial Covering Procedure: LAGRANGIAN-COVER IP Model

exactly 1 group chosen for each Qj

value of 1 contributed to objective function for each triangle covered by a Qj, where that triangle is in a group chosen for that Qj

Variables:

Parameters:

combinatorial covering procedure lagrangian cover ip parameters

T1

T2

T3

T4

T5

Combinatorial Covering Procedure: LAGRANGIAN-COVER IP Parameters

Triangles:

Groups:

Qj’s:

b11=1 b12=0

b21=0b22=1

b31=1b32=1

G1

Q1

G2

Q2

a11=1 a12=1 a13=1

a21=1a22=1a23=1

a31=1a32=0a33=0

a41=1a42=0a43=0

a51=0a52=1a53=0

G3

G3

combinatorial covering procedure lagrangian cover ip constraints

exactly 1 group for each Qj

Combinatorial Covering Procedure: LAGRANGIAN-COVER IP Constraints

k=2

k=1

k=3

b11=1 b12=0

b21=0b22=1

b31=1b32=1

j=1

j=2

Variables:

Parameters:

combinatorial covering procedure lagrangian cover ip constraints27

j=1

j=1

j=1

j=1

j=1

j=2

j=2

j=2

j=2

j=2

Combinatorial Covering Procedure: LAGRANGIAN-COVER IP Constraints

value of 1 contributed to objective function for each triangle covered by a Qj, where that triangle is in a group chosen for that Qj

k=3

k=1

k=2

b11=1 b12=0

b21=0b22=1

b31=1b32=1

a11=1 a12=1 a13=1

a21=1a22=1a23=1

a31=1a32=0a33=0

a41=1a42=0a43=0

a51=0a52=1a53=0

Variables:

Parameters:

combinatorial covering procedure lagrangian cover ip variables

T1

T2

T3

T4

T5

Combinatorial Covering Procedure: LAGRANGIAN-COVER IP Variables

Triangles:

Groups:

Qj’s:

Group choices:

G1 for Q1

G2 for Q2

G1

Q1

g11=1 g12=0g21=0g22=1g31=0g32=0

G2

Q2

t1 , t2=1multiply covered

G3

t1=1 t2=1t3=1t4=1 t5=1

lagrangian relaxation
Lagrangian Relaxation

exactly 1 group chosen for each Qj

value of 1 contributed to objective function for each triangle covered by a Qj, where that triangle is in a group chosen for that Qj

bring into objective function

Variables:

Parameters:

lagrangian relaxation30

1

Lagrangian Relaxation

maximize

Lagrange Multipliers

2

3

4

removing constraints

minimize

2

l>=0 and subtracting term < 0

3

Lagrangian Relaxation LR(l)

1

Lower bounds come from any feasible solution to

1

4

Lagrangian Dual: min LR(l), subject to l >= 0

lagrangian relaxation31
Lagrangian Relaxation

Lagrangian Relaxation LR(l)

LR(l) is separable

SP1

SP2

Solve: if (1-li) >=0

then set ti=1

else set ti=0

Solve: Redistribute:

Solve j sub-subproblems

- compute gkj coefficients

- set to 1 gkjwith largest coefficient

For candidate l values, solve SP1, SP2

lagrangian relaxation32
Lagrangian Relaxation

1

  • Generating lower bound for :
    • SP2 solution yields gkj values feasible for
    • Modify ti values accordingly
    • Result is feasible for

1

1

1

lagrangian relaxation33
Lagrangian Relaxation

SP2

SP1

  • SP1, SP2 have integrality property
    • Solutions unchanged when variable integrality not enforced
    • Optimal value of Lagrangian Dual no better than Linear Programming relaxation of
    • Use as a heuristic:
      • Upper bound for
      • Lower bound for by generating feasible solution to
    • Fast, predictable execution time
    • Optimization software libraries not required

1

1

1

1

lagrangian relaxation34
Lagrangian Relaxation
  • Search l space using subgradient optimization
    • Initialize lis (e.g. 0)
    • Solve SP1 and SP2
    • Update upper bound using sum of SP1, SP2 solutions
    • Generate feasible solution
    • Improve feasible solution using local exchange heuristic
    • Update lower bound using feasible solution
    • Calculate subgradients
    • Calculate step size
    • Take a step in subgradient direction
      • Update lis

Iterate until stopping criteria satisfied