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A Combinatorial Maximum Cover Approach to 2D Translational Geometric Covering Karen Daniels, Arti Mathur, Roger Grinde University of Massachusetts Lowell and University of New Hampshire 11 August, 2003 http://www.cs.uml.edu/~kdaniels Acknowledgment: Cristina Neacsu
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A Family of Covering Problems Q3Q3 Q1Q1 Q2Q2 Sample P and Q P1P1 P2P2 Translated Q Covers P ä Input: ä Covering Items: Q = {Q 1, Q 2,..., Q m } ä Target Items: P = {P 1, P 2,..., P s } ä Subgroup G of Output: solution = { 1, …, j,..., m },, such that Output: a solution = { 1, …, j,..., m },, such that Rigid, 2D, Exact, Polygonal & Point, Translation this work: flexible approximate 3D spline rotation future work: NP-hard
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Sample Application Areas CAD Sensors Lethal Action Locate, Identify, Track, Observe Sensor Coverage Targeting
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COVERINGPROBLEMSCOVERINGPROBLEMS COVERINGPROBLEMSCOVERINGPROBLEMS covering P: finite point sets geometric covering 2D translational covering combinatorial covering P: shapes decomposition: Decomposition with covering partition : VERTEX-COVER, SET-COVER, EDGE-COVER, VLSI logic minimization, facility location covering Polynomial-time algorithms for triangulation and some tilings Q: convex Q: nonconvex BOX-COVER -Thin coverings of the plane with congruent convex shapes -Translational covering of a convex set by a sequence of convex shapes -Translational covering of arbitrary polygonal shapes [CCCG’01] -NP-hard/complete polygon problems -polynomial-time results for restricted orthogonal polygon covering and horizontally convex polygons -approximation algorithms for boundary, corner covers of orthogonal polygons..... Q: identical... 1D interval covered by annuli Source: CCCG’01 Daniels, Inkulu
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Previous Work: CCCG’01 Daniels, Inkulu Q covers P using following constraints: -4 convex pieces of Q -11 points of P -16 constraints: -Q 1 must cover points 1,2,3,4,5 of P -Q’ 2 must cover points 2,6,7,8 of P -Q’ 3 must cover points 5,4,9,10 of P -Q’’ 3 must cover points 4,10,11 of P Q1Q1 Q’ 2 Q’ 3 Q” 3 1 2 3 4 9 8 6 5 7 11 10 P ä Assignments of covering shapes to vertices of target shape constrain positions of covering shapes ä Incremental approach seeks cover with small number of constraints 5 1 2 {1} 3 6 {1}{1} {1, 2} {2}{2} {2}{2} {2}{2} potentially uncovered covered by Q 1 4 Covered by Q 2 {1,2} {1}{1} {2}{2} {2}{2} {2}{2} covered by Q 2 covered by Q 1 1 2 4 3 5 6 covered by Q 1 Covered by Q 2 Convex decomposition of Q leverages convexity coverage property.
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Previous Work: CCCG’01 Daniels, Inkulu Heuristic seeks cover with specified type of intersection graph. 13 {3}1 {1} 12 {3} 6 {1} 5 {3} 4 {2} 3 {2} 2 {1} 11 {2} 10 {2} 9 {3} 8 {1} 7 {3} P Entire approach works well when: - number of vertices of convex hull of P is small; - entire convex hull of P can be covered by Q; - number of faces in convex decomposition of Q is small. 12 13 6 5 4 3 2 1 11 10 9 8 7 Lacks strong mechanism for deciding which Q j ’s should cover which parts of P.
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New Covering Approach Group choices: G1 for Q1 G2 for Q2 T Triangles: T1 T2 T3 Q j ’s:Groups: G1 G2 G3 Q1 Q2 T4 T5
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Minkowski Sum for Containment in ADD-GROUPS Minkowski Sum: Intersection: Containment: QjQj t
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Group Generation Procedure ADD-GROUPS G2 QjQj t 2-contact position removes both x,y degrees of freedom t
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Combinatorial Covering Procedure: LAGRANGIAN-COVER ä ä Integer Programming (IP) formulation maximizes number of triangles covered by selecting one triangle group for each covering shape. ä ä One constraint set is brought into the objective function for Lagrangian Relaxation. ä ä Lagrangian Relaxation is used as a heuristic since optimal value of Lagrangian Dual is no better than Linear Programming relaxation. ä ä Approach was used successfully by Grinde, Daniels (1999) with containment to maximize apparel pattern piece placement.
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Combinatorial Covering Procedure: LAGRANGIAN-COVER IP Parameters Triangles: Q j ’s: Groups: G1 G2 G3 Q1 Q2 G3 T1 T2 T3 T4 T5
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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 T1 T2 T3 T4 T5
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Combinatorial Covering Procedure: LAGRANGIAN-COVER IP Model Variables: Parameters: Brought into objective function for Lagrangian Relaxation Lagrangian Relaxation is used as a heuristic since optimal value of Lagrangian Dual is no better than Linear Programming relaxation. 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
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SUBDIVIDE-TRI Invariant: T is a triangulation of P P uncovered triangle T T’
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Implementation Results Row 4 ALG 1: recent results ALG 2: CCCG’01 Daniels, Inkulu =number of vertices of P #Pts 1,2 = cover description size for ALG 1, 2 Time 1, 2 = run-time in seconds for ALG 1, 2 * Subdivision tolerance of 300 triangles reached ** Run-time cutoff of 10 minutes reached Software Libraries: CGAL, LEDA Row 3 Row 2 Row 13 Row 12 Row 1 Row 10
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Implementation Results Nonconvex Q Polygons Time = 145 seconds # triangles = 35
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Future Work ä Improve triangle subdivision ä Generalize the covering problem Rigid, 2D, Exact, Polygonal & Point, Translation this work: flexible approximate 3D spline rotation future work:
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BACKUP SLIDES
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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
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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
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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
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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
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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
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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
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Lagrangian Relaxation Lagrangian Dual: min LR( ), subject to >= 0 Lagrangian Relaxation LR( ) Lagrange Multipliers 1 2 3 1 2 3 maximize >=0 and subtracting term < 0 removing constraints 4 4 minimize Lower bounds come from any feasible solution to 1
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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
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Lagrangian Relaxation ä Generating lower bound for : ä SP2 solution yields g kj values feasible for ä Modify t i values accordingly ä Result is feasible for 1 1 1 1
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Lagrangian Relaxation ä SP1, SP2 have integrality property ä Solutions unchanged when variable integrality not enforced ä Optimal value of Lagrangian Dual no better than Linear Programming relaxation of ä Use as a heuristic: ä Upper bound for ä Lower bound for by generating feasible solution to ä Fast, predictable execution time ä Optimization software libraries not required SP1 SP2 1 1 1 1
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Lagrangian Relaxation Search space using subgradient optimization Initialize i s (e.g. 0) ä Solve SP1 and SP2 ä Update upper bound using sum of SP1, SP2 solutions ä Generate feasible solution ä Improve feasible solution using local exchange heuristic ä Update lower bound using feasible solution ä Calculate subgradients ä Calculate step size ä Take a step in subgradient direction Update i s Iterate until stopping criteria satisfied
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