7/25/ 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

–Sensor coverage: –Repair work: collection of pieces cover a hole Motivation for 2D Polygonal Covering Supported under NSF/DARPA CARGO program NP-hard problem

Box Covering –Goal: Translate a collection of boxes (orthotopes) Q = { Q 1, Q 2,..., Q N } to cover another box P in 2 d, 3 d, … –Motivation: Boxes can form enclosures for general shapes. With Masters student B. England Partial cover (red part uncovered) Full cover 2 d views of 3 d covering Supported under NSF/DARPA CARGO program NP-hard problem 1 st published results in > 2 d d = dimension 20 covering shapes 40 covering shapes

covering P: finite point sets geometric covering translational covering combinatorial covering P: shapes decomposition: decomposition with covering partition: VERTEX-COVER, SET-COVER (including [Gri99]), EDGE-COVER, VLSI logic minimization, facility location Polynomial-time algorithms for triangulation [Cha91] and some tilings Q: convex Q: nonconvex BOX-COVER [Fow81] -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..... 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] Q: identical... 1D interval covered by annuli using approximation algorithm [Hoc87] NP-complete Selected Prior Covering Work

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 Based on Daniels and Grinde, IIE Transactions, 1999 IP model, maximizing number of parts covered, treated with Lagrangian Heuristic + 1-OPT parts g 11 = {1,3} g 12 = {2,4} g 21 = {1,2} g 22 = {3,4} g 31 = {1} g 32 = {2} g 33 = {3} g 34 = {4} C 1 = {g 11,g 12 } C 2 = {g 21,g 22 } C 3 = {g 31,g 32,g 33,g 34 } Problem: choose just one part group g jk from the set of part groups C j for each cover shape Q j such that every part of P is in one of the chosen part groups. Solution: {g 11,g 21,g 34 }

Key Volume Expressions volume quantized volume quantized effective volume effective volume  = a generic part of P

N = number of covering shapes d = dimension LGC_Cover( ) = modified Lagrangian Heuristic + 1-OPT  = total number of parts of P Uniform refinement scheme, unlike general polygonal approach of [Dan03], which subdivides one triangle during each iteration of repeat loop.  = a generic part of P Heuristic:

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: cover shape volume= = 1.21 Minkowski sum of two sets A and B is 4 identical square covering shapes Daniels, et al. [CCCG2003]: 167 triangular parts 888 groups 94 triangle vertices 875 seconds 450 MHz CPU Sun SPARC Ultra 60 TM with 512 MB memory Current paper: 4 square parts 5 square vertices 80 groups 0.2 seconds

Experimental Highlights Results in 3 d and 4 d : N = number of covering shapes d = dimension  = total number of parts of P csv = cover shape volume 3 GHz 64-bit Intel Pentium TM D CPU with 2 GB memory maximum aspect ratio of a covering shape = 4

Experimental Highlights More results in 3 d : 2 views of 3 d covering using OpenGL 2 d views of 3 d covering using OpenGL 1 GHz Intel Pentium TM 4 CPU with 1/2 GB memory d N  #groups time (seconds) % covered 3 204,09647,603,138< ,0962,344,940<

Problem Instance “Hardness” Characterization N = number of covering shapes d = dimension  = total number of parts of P Dimension-independent Volume MarginQuantized Effective Volume Ratio  = a generic part of P

tp = total points. r = correlation coefficient between  and % coverage 1 GHz Intel Pentium TM 4 CPU with 1/2 GB memory ~10 instances for each parameter combination Effectiveness of 

3 GHz 64-bit Intel Pentium TM D CPU with 2 GB memory test added d #  N # calls % savings instances saved Effectiveness of  100% coverage was reached by original & revised heuristic in all these cases. # calls saved = average per instance % savings = average relative % savings of LCG_Cover( ) calls Heuristic:

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 2 d instances 87% of 3 d instances 64% of 4 d instances Test suite = subset of 30 of our randomly generated instances: 10 2 d, 10 3d, 10 4d 3 GHz 64-bit Intel Pentium TM D CPU with 2 GB memory 1-OPT heuristic swaps groups for cover shape that best improves objective function until no improvement.

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 2 d, 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%) 1-OPT heuristic swaps groups for cover shape that best improves objective function until no improvement. Test suite = subset of 30 of our randomly generated instances: 10 2 d, 10 3d, 10 4d 3 GHz 64-bit Intel Pentium TM D CPU with 2 GB memory monotonically increasing

Conclusion & Future Work Partition-based, translational, box covering heuristic has dimension as an input. Set covering approach uses uniform refinement. First 3 d, 4 d 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 2 d 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 2 d general covering heuristic treat 3 d general covering –Allow rotations

18 References Acknowledgement: Thanks to Michelle Daniels for comments.

19 For More Information Web sites: Thanks for your attention ! Questions?

BACKUP SLIDES (from CCCG 2003, etc.)

With graduate students R. Inkulu, A. Mathur, C.Neacsu, & UNH professor R. Grinde 2D Polygonal Covering [CCCG 2001,CCCG2003] Q3Q3 Q1Q1 Q2Q2 Sample P and Q P1P1 P2P2 Translated Q Covers P P1P1 Q1Q1 Q2Q2 Q3Q3 P2P2 Translational 2D Polygon Covering Input: –Covering polygons Q = { Q 1, Q 2,..., Q m } –Target polygons (or point-sets) P = { P 1, P 2,..., P n } Output: –Translations  = {  1,  2,...,  m } such that Supported under NSF/DARPA CARGO program

2D B-Spline Covering [CORS/INFORMS2004, UMass Lowell Student Research Symposium 2004, Computers Graphics Forum, 2006] With graduate student C. Neacsu Supported under NSF/DARPA CARGO program NP-hard problem motivated by 3D CAD

Covering Web Site With graduate student C. Neacsu and undergraduate A. Hussin

Combinatorial Covering Procedure: LAGRANGIAN-COVER IP Model Variables: Parameters: exactly 1 group chosen for each Q j value of 1 contributed to objective function for each triangle covered by a Q j, where that triangle is in a group chosen for that Q j

Combinatorial Covering Procedure: LAGRANGIAN-COVER IP Parameters Triangles: Q j ’s: Groups: G1 G2 G3 Q1 Q2 G3 a 11 =1 a 12 =1 a 13 =1 a 21 =1 a 22 =1 a 23 =1 a 31 =1 a 32 =0 a 33 =0 a 41 =1 a 42 =0 a 43 =0 a 51 =0 a 52 =1 a 53 =0 b 11 =1 b 12 =0 b 21 =0 b 22 =1 b 31 =1 b 32 =1 T1 T2 T3 T4 T5

Combinatorial Covering Procedure: LAGRANGIAN-COVER IP Constraints Variables: Parameters: k=1 k=2 k=3 j=1 j=2 b 11 =1 b 12 =0 b 21 =0 b 22 =1 b 31 =1 b 32 =1 exactly 1 group for each Q j

Combinatorial Covering Procedure: LAGRANGIAN-COVER IP Constraints Variables: Parameters: value of 1 contributed to objective function for each triangle covered by a Q j, where that triangle is in a group chosen for that Q j k=1 k=2 k=3 j=1 j=2 b 11 =1 b 12 =0 b 21 =0 b 22 =1 b 31 =1 b 32 =1 j=1 j=2 j=1 j=2 j=1 j=2 j=1 j=2 a 11 =1 a 12 =1 a 13 =1 a 21 =1 a 22 =1 a 23 =1 a 31 =1 a 32 =0 a 33 =0 a 41 =1 a 42 =0 a 43 =0 a 51 =0 a 52 =1 a 53 =0

Combinatorial Covering Procedure: LAGRANGIAN-COVER IP Variables Triangles: Q j ’s: Groups: G1 G2 G3 Q1 Q2 Group choices: G1 for Q1 G2 for Q2 g 11 =1 g 12 =0 g 21 =0 g 22 =1 g 31 =0 g 32 =0 t 1 =1 t 2 =1 t 3 =1 t 4 =1 t 5 =1 t 1, t 2 =1 multiply covered T1 T2 T3 T4 T5

Lagrangian Relaxation Variables: Parameters: bring into objective function exactly 1 group chosen for each Q j value of 1 contributed to objective function for each triangle covered by a Q j, where that triangle is in a group chosen for that Q j

Lagrangian Relaxation Lagrangian Dual: min LR( ), subject to >= 0 Lagrangian Relaxation LR( ) Lagrange Multipliers maximize >=0 and subtracting term < 0 removing constraints 4 4 minimize Lower bounds come from any feasible solution to 1

Lagrangian Relaxation Lagrangian Relaxation LR( ) LR( ) is separable SP1 SP2 Solve: if (1- i ) >=0 then set t i =1 else set t i =0 Solve: Redistribute: Solve j sub-subproblems - compute g kj coefficients - set to 1 g kj with largest coefficient For candidate values, solve SP1, SP2

Lagrangian Relaxation Generating lower bound for : –SP2 solution yields g kj values feasible for –Modify t i values accordingly –Result is feasible for

Lagrangian Relaxation 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 SP1 SP

Lagrangian Relaxation Search  space using subgradient optimization –Initialize i s (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 i s Iterate until stopping criteria satisfied