A Combinatorial Maximum Cover Approach to 2D Translational Geometric Covering

1
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

• Input
• Covering Items Q Q1, Q2 , ... , Qm
• Target Items P P1, P2 , ... , Ps
• Subgroup G of
• Output a solution g g 1, ,g j , ... , g m,
  , such that

Translated Q Covers P
Sample P and Q
P2
P1
Q1
Q2
Q3
NP-hard
Rigid, 2D, Exact, Polygonal Point, Translation
this work
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Sample Application Areas
Sensors
Locate, Identify, Track, Observe
Lethal Action
CAD
Sensor Coverage Targeting
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COVERINGPROBLEMS
Source CCCG01 Daniels, Inkulu
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Previous Work CCCG01 Daniels, Inkulu

• Assignments of covering shapes to vertices of
  target shape constrain positions of covering
  shapes
• Incremental approach seeks cover with small
  number of constraints

• Q covers P using following constraints
• 4 convex pieces of Q
• 11 points of P
• 16 constraints
• Q1 must cover points 1,2,3,4,5 of P
• Q2 must cover points 2,6,7,8 of P
• Q3 must cover points 5,4,9,10 of P
• Q3 must cover points 4,10,11 of P

Convex decomposition of Q leverages convexity
coverage property.
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Previous Work CCCG01 Daniels, Inkulu
Heuristic seeks cover with specified type of
intersection graph.

• 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.

Lacks strong mechanism for deciding which Qjs
should cover which parts of P.
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New Covering Approach
T
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Minkowski Sum for Containment in ADD-GROUPS
Minkowski Sum
Intersection
Containment
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Group Generation ProcedureADD-GROUPS
2-contact position removes both x,y degrees of
freedom
t
G2
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Combinatorial Covering Procedure LAGRANGIAN-COVER

• Integer Programming (IP) formulation maximizes
  number of triangles covered by selecting one
  triangle group for each covering shape.
• One constraint set is brought into the 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.
• 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
Groups
Qjs
G1
Q1
G2
Q2
G3
G3
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Combinatorial Covering Procedure
LAGRANGIAN-COVER IP Variables
Triangles
Groups
Qjs
Group choices G1 for Q1 G2 for Q2
G1
Q1
G2
Q2
G3
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Combinatorial Covering Procedure
LAGRANGIAN-COVER IP Model
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 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
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SUBDIVIDE-TRI
Invariant T is a triangulation of P
T
T
uncovered triangle
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Implementation Results
ALG 1 recent results ALG 2 CCCG01
Daniels, Inkulu hnumber 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
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Implementation Results
Nonconvex Q Polygons
triangles 35
Time 145 seconds
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Future Work

• Improve triangle subdivision
• Generalize the covering problem

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BACKUP SLIDES
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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
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Combinatorial Covering Procedure
LAGRANGIAN-COVER IP Parameters
Triangles
Groups
Qjs
b111 b120 b210 b221 b311 b321
G1
Q1
G2
Q2
a111 a121 a131 a211 a221 a231
a311 a320 a330 a411 a420
a430 a510 a521 a530
G3
G3
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Combinatorial Covering Procedure
LAGRANGIAN-COVER IP Constraints
k2
k1
k3
b111 b120 b210 b221 b311 b321
j1
j2
Variables
Parameters
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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
k3
k1
k2
b111 b120 b210 b221 b311 b321
a111 a121 a131 a211 a221 a231
a311 a320 a330 a411 a420
a430 a510 a521 a530
Variables
Parameters
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Combinatorial Covering Procedure
LAGRANGIAN-COVER IP Variables
Triangles
Groups
Qjs
Group choices G1 for Q1 G2 for Q2
G1
Q1
g111 g120 g210 g221 g310 g320
G2
Q2
t1 , t21 multiply covered
G3
t11 t21 t31 t41 t51
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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
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Lagrangian Relaxation
1
Lagrange Multipliers
maximize
2
3
4
removing constraints
minimize
2
lgt0 and subtracting term lt 0
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Lagrangian Relaxation LR(l)
1
Lower bounds come from any feasible solution to
1
4
Lagrangian Dual min LR(l), subject to l gt 0
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Lagrangian Relaxation
Lagrangian Relaxation LR(l)
LR(l) is separable
SP1
SP2
Solve if (1-li) gt0 then set ti1 else set ti0
Solve Redistribute Solve j
sub-subproblems - compute gkj
coefficients - set to 1 gkj with
largest coefficient
For candidate l values, solve SP1, SP2
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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
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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
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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