A Combinatorial Maximum Cover Approach to 2D Translational

Geometric Covering

A Combinatorial Maximum Cover Approach to 2D Translational

Geometric Covering

Karen Daniels, Arti Mathur, Roger GrindeKaren Daniels, Arti Mathur, Roger GrindeUniversity of Massachusetts Lowell and University of New HampshireUniversity of Massachusetts Lowell and University of New Hampshire

11 August, 200311 August, 2003http://www.cs.uml.edu/~kdaniels

Acknowledgment: Cristina Neacsu

A Family of Covering Problems A Family of Covering Problems

Q3

Q1 Q2

Sample P and Q

P1

P2

Translated Q Covers P

Input: Input: Covering Items: Covering Items: QQ = { = {QQ11, , QQ2 2 , ... , , ... , QQmm}}

Target Items: Target Items: PP = { = {PP11, , PP2 2 , ... , , ... , PPss}}

Subgroup Subgroup GG of of Output: Output: a solution solution = { = { 11, …,, …, j j , ... , , ... , mm}, , such that}, , such that

mjj j

QP

1

)()( nRIso

Rigid, 2D, Exact, Polygonal & Point, Translation

Gj

this work:

flexible approximate3D spline rotationfuture work:

NP-hard

Sample Application AreasSample Application Areas

CADCAD

Sensors

Lethal ActionLocate, Identify,Track, Observe

Sensor CoverageSensor CoverageTargetingTargeting

COVERING

PROBLEMS

COVERING

PROBLEMS

covering

P: finite point sets

geometric covering

2D translational covering

combinatorial covering

P: shapes

decomposition:

Decomposition with covering )()( iijj QoverlapmayQ

partition:

)()( iijj QoverlapcannotQ

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

. . . .

. . . . . .

mj

j jQP

1

)(

mj

j jQP

1

)(

Q: identical

. . . 1D interval covered by annuli

Source: CCCG’01 Daniels, Inkulu

Previous Work: CCCG’01 Daniels, InkuluPrevious Work: CCCG’01 Daniels, Inkulu

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-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 PQ1

Q’2

Q’3

Q”3

1

2

3

49

8

65

7

11

10

P

Assignments of Assignments of covering shapes to covering shapes to vertices of target shape vertices of target shape constrain positions of constrain positions of covering shapescovering shapes

Incremental approach Incremental approach seeks cover with small seeks cover with small number of constraintsnumber of constraints

5

1

2 {1}

3

6

{1}

{1, 2}

{2}

{2}

{2}potentially uncovered

covered by Q1

4

Covered by Q2

{1,2}

{1}

{1, 2}

{2}

{2}

{2}covered by Q2

covered by Q1

1

2

4

35

6covered by Q1

Covered by Q2

Convex decomposition of Q leverages convexity coverage property.

Previous Work: CCCG’01 Daniels, InkuluPrevious 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

21

11

10

9

8 7

Lacks strong mechanism for deciding which Qj’s should cover which parts of P.

New Covering ApproachNew Covering Approach

Group choices:

G1 for Q1G2 for Q2

T

Triangles:

T1

T2

T3

Qj’s:Groups:

G1

G2

G3

Q1

Q2T4

T5

Minkowski Sum for Containment in ADD-GROUPSMinkowski Sum for Containment in ADD-GROUPS

)()( inside jj QtvQvt

jj QqttqtQt ,'|'

)()( jj QtvQvt

Minkowski Sum:

Intersection:

Containment:

jQt

Qjt

)( jQt

jQt

Group Generation ProcedureADD-GROUPSGroup Generation ProcedureADD-GROUPS

G2

Qjt

2-contact position removes both x,y degrees of freedom

t

K

Combinatorial Covering Procedure: LAGRANGIAN-COVERCombinatorial 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.

Combinatorial Covering Procedure: LAGRANGIAN-COVER IP ParametersCombinatorial Covering Procedure: LAGRANGIAN-COVER IP Parameters

Triangles: Qj’s:Groups:

G1

G2

G3

Q1

Q2G3

T1

T2

T3

T4

T5

Combinatorial Covering Procedure: LAGRANGIAN-COVER IP VariablesCombinatorial Covering Procedure: LAGRANGIAN-COVER IP Variables

Triangles: Qj’s:Groups:

G1

G2

G3

Q1

Q2

Group choices:

G1 for Q1G2 for Q2T1

T2

T3

T4

T5

Combinatorial Covering Procedure: LAGRANGIAN-COVER IP ModelCombinatorial 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 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

kjg

SUBDIVIDE-TRISUBDIVIDE-TRI

Invariant: T is a triangulation of P

P

uncovered triangle

T T’

Implementation ResultsImplementation 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: Software Libraries: CGAL, LEDACGAL, LEDARow 3

Row 2

Row 13

Row 12

Row 1

Row 10

Implementation ResultsImplementation Results

Nonconvex Q Polygons

Time = 145 seconds # triangles = 35

Future WorkFuture Work

Improve triangle subdivisionImprove triangle subdivision Generalize the covering problemGeneralize the covering problem

Rigid, 2D, Exact, Polygonal & Point, Translationthis work:

flexible approximate3D spline rotationfuture work:

BACKUP SLIDESBACKUP SLIDES

Combinatorial Covering Procedure: LAGRANGIAN-COVER IP ModelCombinatorial Covering Procedure: LAGRANGIAN-COVER IP Model

Variables: Parameters:

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

kjg

Combinatorial Covering Procedure: LAGRANGIAN-COVER IP ParametersCombinatorial Covering Procedure: LAGRANGIAN-COVER IP Parameters

Triangles: Qj’s:Groups:

G1

G2

G3

Q1

Q2G3

a11=1 a12=1 a13=1

a21=1 a22=1 a23=1

a31=1 a32=0 a33=0

a41=1 a42=0 a43=0

a51=0 a52=1 a53=0

b11=1 b12=0

b21=0 b22=1

b31=1 b32=1

T1

T2

T3

T4

T5

Combinatorial Covering Procedure: LAGRANGIAN-COVER IP ConstraintsCombinatorial Covering Procedure: LAGRANGIAN-COVER IP Constraints

1312111 ggg

1322212 ggg

13111 gg

Variables: Parameters:

k=1 k=2 k=3

j=1

j=2

b11=1 b12=0

b21=0 b22=1

b31=1 b32=1 13222 gg

exactly 1 group for each Qjkjg

Combinatorial Covering Procedure: LAGRANGIAN-COVER IP ConstraintsCombinatorial Covering Procedure: LAGRANGIAN-COVER IP Constraints

321322121211

3113211211111

gagaga

gagagat

322322221221

3123212211212

gagaga

gagagat

323322321231

3133213211313

gagaga

gagagat

Variables: Parameters:

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

325322521251

3153215211515

gagaga

gagagat

k=1 k=2 k=3

j=1

j=2322231111 ggggt

322231112 ggggt

113 gt

225 gt

b11=1 b12=0

b21=0 b22=1

b31=1 b32=1

324322421241

3143214211414

gagaga

gagagat

114 gt

j=1

j=2

j=1

j=2

j=1

j=2

j=1

j=2

a11=1 a12=1 a13=1

a21=1 a22=1 a23=1

a31=1 a32=0 a33=0

a41=1 a42=0 a43=0

a51=0 a52=1 a53=0

Combinatorial Covering Procedure: LAGRANGIAN-COVER IP VariablesCombinatorial Covering Procedure: LAGRANGIAN-COVER IP Variables

Triangles: Qj’s:Groups:

G1

G2

G3

Q1

Q2

Group choices:

G1 for Q1G2 for Q2

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

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

21322231112 ggggt

113 gt

322231111 ggggt 21

225 gt

t1 , t2=1 multiply covered

11311 gg

12322 gg

11

11

T1

T2

T3

T4

T5

114 gt 11

11

11

Lagrangian RelaxationLagrangian Relaxation

Variables: Parameters:

bring into objective function

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

kjg

Lagrangian RelaxationLagrangian Relaxation

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

Lagrangian Relaxation LR()

T

i

m

j bkkjikii

kj

gat1 1 1

T

i

m

j bkkjikii

kj

gat1 1 1

Lagrange Multipliers

1

2

3

1

2

3

maximize

>=0 and subtracting term < 0

removing constraints

4

4minimize

Lower bounds come from any feasible solution to 1

kjg

kjg

kjg

Lagrangian RelaxationLagrangian Relaxation

Lagrangian Relaxation LR()

T

i

m

j bkkjikii

kj

gat1 1 1

i

T

ii t

1

1 maximize

T

i

m

j bkkjiki

kj

ga1 1 1

LR() is separable

SP1 SP2

Solve: if (1-i) >=0

then set ti=1

else set ti=0Solve: Redistribute:

Solve j sub-subproblems

- compute gkj coefficients

- set to 1 gkj with largest coefficient

m

j bk

T

ikjiki

kj

ga1 1 1

For candidate values, solve SP1, SP2

kjg

kjg

Lagrangian RelaxationLagrangian Relaxation

Generating lower bound for :Generating lower bound for : SP2 solution yields SP2 solution yields ggkjkj values feasible for values feasible for

Modify tModify tii values accordingly values accordingly

Result is feasible for Result is feasible for

1

1

1

1

kjg

Lagrangian RelaxationLagrangian Relaxation

SP1, SP2 have SP1, SP2 have integralityintegrality property property Solutions unchanged when variable integrality not enforcedSolutions unchanged when variable integrality not enforced Optimal value of Lagrangian Dual no better than Linear Programming relaxation Optimal value of Lagrangian Dual no better than Linear Programming relaxation

of of Use as a heuristic:Use as a heuristic:

Upper bound for Upper bound for Lower bound for by generating feasible solution toLower bound for by generating feasible solution to

Fast, predictable execution timeFast, predictable execution time Optimization software libraries not requiredOptimization software libraries not required

i

T

ii t

1

1 maximize

T

i

m

j bkkjiki

kj

ga1 1 1

SP1

SP2

1

1

1

kjg

1

Lagrangian RelaxationLagrangian Relaxation

Search Search space using space using subgradient optimizationsubgradient optimization Initialize Initialize iis (e.g. 0)s (e.g. 0) Solve SP1 and SP2Solve SP1 and SP2 Update upper bound using sum of SP1, SP2 solutionsUpdate upper bound using sum of SP1, SP2 solutions Generate feasible solution Generate feasible solution Improve feasible solution using local exchange heuristicImprove feasible solution using local exchange heuristic Update lower bound using feasible solutionUpdate lower bound using feasible solution Calculate subgradientsCalculate subgradients Calculate step sizeCalculate step size Take a step in subgradient directionTake a step in subgradient direction

Update Update iiss

Iterate until stopping criteria satisfied