Irregular Polyomino Tiling via Integer Programming
Oleg Prokopyev Serdar Karademir
Department of Industrial Engineering University of Pittsburgh
AFOSR Optimization and Discrete Mathematics Program Review
April 18, 2012
PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds Polyominoes: , ,... F1 F2
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PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds Polyomino Tiling and Computational Complexity
Polyomino Tiling Problem Tiling a finite rectangular region with a given finite set of polyominoes from the same family with no restriction on the number of copies of each polyomino used.
Various versions of the polyomino tiling problem are known to be NP -hard: Moore and Robson, 2001; Demaine and Demaine, 2007
PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds Polyomino Tiling and Computational Complexity
Polyomino Tiling Problem Tiling a finite rectangular region with a given finite set of polyominoes from the same family with no restriction on the number of copies of each polyomino used.
Various versions of the polyomino tiling problem are known to be NP -hard: Moore and Robson, 2001; Demaine and Demaine, 2007
PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds Polyomino Tiling and Computational Complexity
Polyomino Tiling Problem Tiling a finite rectangular region with a given finite set of polyominoes from the same family with no restriction on the number of copies of each polyomino used.
Integer Knapsack Feasibility
Given nonnegative integers s1, s2, . . ., sn, and K, we need to check whether there exist nonnegative integers x1,x2,...,xn satisfying s1x1 + s2x2 + . . . + snxn = K . (⋆)
Reduction
Consider a set of rectangular polyominoes 1 s ,..., 1 sn. P × 1 × Then (⋆) has an integral solution iff a rectangular 1 K strip can × be exactly covered (tiled) by some polyominoes from . P PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds Polyomino Tiling and Computational Complexity
Polyomino Tiling Problem Tiling a finite rectangular region with a given finite set of polyominoes from the same family with no restriction on the number of copies of each polyomino used.
Integer Knapsack Feasibility
Given nonnegative integers s1, s2, . . ., sn, and K, we need to check whether there exist nonnegative integers x1,x2,...,xn satisfying s1x1 + s2x2 + . . . + snxn = K . (⋆)
Reduction
Consider a set of rectangular polyominoes 1 s ,..., 1 sn. P × 1 × Then (⋆) has an integral solution iff a rectangular 1 K strip can × be exactly covered (tiled) by some polyominoes from . P PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds Phased Array Antennas
Phased array antennas are composed of many stationary antenna elements
(b) Main beam is electronically steered using phase shift and time delay A wide range of applications
PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds Polyominoes in Antenna Design Ideally, one would like to have controls at the element level However, too many controls is expensive to implement A group of elements is used to form a subarray which is treated as an oversized element Sidelobes (beams with a direction other than the main one) occur due to periodicity introduced by identical rectangular subarrays Irregularly tiled polyomino-shaped subarrays result in a major decrease in sidelobes [Mailloux et al., 2009]
PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds Polyominoes in Antenna Design Ideally, one would like to have controls at the element level However, too many controls is expensive to implement A group of elements is used to form a subarray which is treated as an oversized element Sidelobes (beams with a direction other than the main one) occur due to periodicity introduced by identical rectangular subarrays Irregularly tiled polyomino-shaped subarrays result in a major decrease in sidelobes [Mailloux et al., 2009]
PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds Polyominoes in Antenna Design Ideally, one would like to have controls at the element level However, too many controls is expensive to implement A group of elements is used to form a subarray which is treated as an oversized element Sidelobes (beams with a direction other than the main one) occur due to periodicity introduced by identical rectangular subarrays Irregularly tiled polyomino-shaped subarrays result in a major decrease in sidelobes [Mailloux et al., 2009].
0 0 0 −20 −20 −20 −40 −40 −40 −60 −60 −60 −80 −80 −80 1 1 1 0.5 1 0.5 1 0.5 1 0.5 0.5 0.5 0 0 0 0 0 0 −0.5 −0.5 −0.5 −0.5 −0.5 −0.5 −1 −1 −1 −1 −1 −1 (a) (b) (c) PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds
1 1 −5 −5 0.8 0.8 −10 −10 0.6 0.6 −15 −15 0.4 0.4 −20 −20 0.2 0.2 − −25 y v 0 25 y v 0 − − −0.2 30 −0.2 30 − − −0.4 35 −0.4 35 −0.6 −40 −0.6 −40 −0.8 −45 −0.8 −45 −1 −50 −1 −50 x −1 −0.6 −0.2 0 .2 0 .6 1 x −1 −0.6 −0.2 0 .2 0 .6 1 u u (a) Array of rectangular subarrays (c) Array of octomino subarrays. Subarray phase centers are indicated by red dots
1 1 −5 −5 0.8 0.8 −10 −10 0.6 0.6 −15 −15 0.4 0.4 −20 −20 0.2 0.2 − − y v 0 25 y v 0 25 − − −0.2 30 −0.2 30 − −0.4 35 −0.4 −35 −0.6 −40 −0.6 −40 −0.8 −45 −0.8 −45 −1 −50 −1 −50 x −1 −0.6 −0.2 0 .2 0 .6 1 x −1 −0.6 −0.2 0 .2 0 .6 1 u u (b) Array of omnidirectional elements at rectangular subarray phase centers (d) Array of omnidirectional elements at octomino subarray phase centers
[Mailloux et al., 2009]
PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds A Set Theoretic Description of Polyomino Tiling Problem
! " # $ % &
! '!(") ℓ - the family of polyominoes F with ℓ Z+ squares " '"(!) '"(") ∈
# '#(") '#(#) The rectangle hull of a polyomino - the smallest rectangular box that $ the polyomino would fit in
% f ℓ - rectangle hull located at & ∈ F (0, 0): f = (i , j ),..., (iℓ, jℓ) { 1 1 }
PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds A Set Theoretic Description of Polyomino Tiling Problem
The board - = (i, j) : 0 i< ! !"# !"$ !"% B { ≤ m; 0 j & '&(!"#) 1 K ! = f ,...,f Fℓ to tile P { }⊆ B &"# '&"#(!) '&"#(!"#) k k fpq = (i + p, j + q) : (i, j) f &"$ '&"$(!"#) '&"$(!"$) { ∈ } &"% The tiling problem ( , ) B P Given and , find an exact cover of B P B using subsets f k : f k ; f k . { pq pq ⊆B ∈ P} PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds A Set Theoretic Description of Polyomino Tiling Problem The board - = (i, j) : 0 i< ! !"# !"$ !"% B { ≤ m; 0 j & '&(!"#) 1 K ! = f ,...,f Fℓ to tile P { }⊆ B &"# '&"#(!) '&"#(!"#) k k fpq = (i + p, j + q) : (i, j) f &"$ '&"$(!"#) '&"$(!"$) { ∈ } &"% The tiling problem ( , ) B P Given and , find an exact cover of B P B using subsets f k : f k ; f k . { pq pq ⊆B ∈ P} PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds Exact Covering Formulation k k For (i, j) and f : Iij = (k,p,q) (i, j) f ∈B pq ⊆B { | ∈ pq} k k Associating 0–1 variable xpq with the set fpq: xk = 1 (i, j) X pq ∀ ∈B (kpq)∈Iij PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds Exact Covering Formulation k k For (i, j) and f : Iij = (k,p,q) (i, j) f ∈B pq ⊆B { | ∈ pq} k k Associating 0–1 variable xpq with the set fpq: xk = 1 (i, j) X pq ∀ ∈B (kpq)∈Iij PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds 1 1 −5 −5 0.8 0.8 −10 −10 0.6 0.6 −15 −15 0.4 0.4 −20 −20 0.2 0.2 − −25 y v 0 25 y v 0 − − −0.2 30 −0.2 30 − − −0.4 35 −0.4 35 −0.6 −40 −0.6 −40 −0.8 −45 −0.8 −45 −1 −50 −1 −50 x −1 −0.6 −0.2 0 .2 0 .6 1 x −1 −0.6 −0.2 0 .2 0 .6 1 u u (a) Array of rectangular subarrays (c) Array of octomino subarrays. Subarray phase centers are indicated by red dots 1 1 −5 −5 0.8 0.8 −10 −10 0.6 0.6 −15 −15 0.4 0.4 −20 −20 0.2 0.2 − − y v 0 25 y v 0 25 − − −0.2 30 −0.2 30 − −0.4 35 −0.4 −35 −0.6 −40 −0.6 −40 −0.8 −45 −0.8 −45 −1 −50 −1 −50 x −1 −0.6 −0.2 0 .2 0 .6 1 x −1 −0.6 −0.2 0 .2 0 .6 1 u u (b) Array of omnidirectional elements at rectangular subarray phase centers (d) Array of omnidirectional elements at octomino subarray phase centers [Mailloux et al., 2009] IDEA Use subarray phase centers PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds 1 1 −5 −5 0.8 0.8 −10 −10 0.6 0.6 −15 −15 0.4 0.4 −20 −20 0.2 0.2 − −25 y v 0 25 y v 0 − − −0.2 30 −0.2 30 − − −0.4 35 −0.4 35 −0.6 −40 −0.6 −40 −0.8 −45 −0.8 −45 −1 −50 −1 −50 x −1 −0.6 −0.2 0 .2 0 .6 1 x −1 −0.6 −0.2 0 .2 0 .6 1 u u (a) Array of rectangular subarrays (c) Array of octomino subarrays. Subarray phase centers are indicated by red dots 1 1 −5 −5 0.8 0.8 −10 −10 0.6 0.6 −15 −15 0.4 0.4 −20 −20 0.2 0.2 − − y v 0 25 y v 0 25 − − −0.2 30 −0.2 30 − −0.4 35 −0.4 −35 −0.6 −40 −0.6 −40 −0.8 −45 −0.8 −45 −1 −50 −1 −50 x −1 −0.6 −0.2 0 .2 0 .6 1 x −1 −0.6 −0.2 0 .2 0 .6 1 u u (b) Array of omnidirectional elements at rectangular subarray phase centers (d) Array of omnidirectional elements at octomino subarray phase centers [Mailloux et al., 2009] IDEA Use subarray phase centers PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds Measure of Irregularity Designate one square of the polyomino as its center of gravity Definition of the center of gravity is very flexible; for the phased array antenna case, it may correspond to the subarray phase center Regular Tilings Centers of gravity are expected to accumulate on certain rows and columns of the board due to periodic patterns PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds Measure of Irregularity Designate one square of the polyomino as its center of gravity Definition of the center of gravity is very flexible; for the phased array antenna case, it may correspond to the subarray phase center Regular Tilings Centers of gravity are expected to accumulate on certain rows and columns of the board due to periodic patterns PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds Information Theoretic Entropy X - a discrete random variable with finite support pi - probability of outcome i H(X) - the information theoretic entropy of X: H(X)= pi lg(pi) − X i Observation Uniform distribution has the maximum entropy PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds Information Theoretic Entropy X - a discrete random variable with finite support pi - probability of outcome i H(X) - the information theoretic entropy of X: H(X)= pi lg(pi) − X i Observation Uniform distribution has the maximum entropy PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds Information Theoretic Entropy X - a discrete random variable with finite support pi - probability of outcome i H(X) - the information theoretic entropy of X: H(X)= pi lg(pi) − X i Observation Uniform distribution has the maximum entropy PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds Probability Distribution of Centers of Gravity For a given tiling : T ri (cj) - number of centers of gravity on row i (column j) T - total number of polyominoes used by T ri cj 2T ( 2T ) - row (column) probabilities Interpreting probabilities If a center of gravity on the board is chosen at random, what is the probability of finding it on a certain row (column) PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds Probability Distribution of Centers of Gravity For a given tiling : T ri (cj) - number of centers of gravity on row i (column j) T - total number of polyominoes used by T ri cj 2T ( 2T ) - row (column) probabilities Interpreting probabilities If a center of gravity on the board is chosen at random, what is the probability of finding it on a certain row (column) PITT Minimize entropy: H(X)=2.8842 Maximize entropy: H(X)=3.6889 Motivation Optimization Model Exact Methods Heuristics Approximation Bounds Completing the Formulation k k gpq - the center of gravity for fpq k Ri = (k,p,q) j such that (i, j)= g { |∃ pq} k Cj = (k,p,q) i such that (i, j)= g { |∃ pq} Nonlinear Mixed Binary Program r r c c P : max − i lg i − j lg j NL X 2T 2T X 2T 2T i j s.t. xk = 1 ∀ (i, j) ∈ B X pq (kpq)∈Iij k ri = x ∀ i X pq (kpq)∈Ri k cj = x ∀ j X pq (kpq)∈Cj k xpq ∈ {0, 1}, ri, cj ≥ 0 ∀ i,j,p,q,k PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds Completing the Formulation k k gpq - the center of gravity for fpq k Ri = (k,p,q) j such that (i, j)= g { |∃ pq} k Cj = (k,p,q) i such that (i, j)= g { |∃ pq} Nonlinear Mixed Binary Program r r c c P : max − i lg i − j lg j NL X 2T 2T X 2T 2T i j s.t. xk = 1 ∀ (i, j) ∈ B X pq (kpq)∈Iij k ri = x ∀ i X pq (kpq)∈Ri k cj = x ∀ j X pq (kpq)∈Cj k xpq ∈ {0, 1}, ri, cj ≥ 0 ∀ i,j,p,q,k PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds Linearization m n board and polyomino family ℓ × B F mn T = ℓ centers of gravity on the board ri and cj replaced with 2(T + 1) new binary variables: ri0, ri1,...,riT { } cj0,cj1,...,cjT { } PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds Linear Mixed Binary Program m T t t n T t t PL: max − lg rit − lg cjt 2T 2T 2T 2T Xi=1 Xt=1 jX=1 Xt=1 k s.t. xpq = 1 ∀ (i,j) ∈ B (kpqX)∈Iij T T k t rit = xpq , rit = 1 ∀i Xt=1 (kpqX)∈Ri Xt=0 T T k t cjt = xpq, cjt = 1 ∀ j Xt=1 (kpqX)∈Cj Xt=0 k xpq, rit, cjt ∈ {0, 1} ∀ i,j,p,q,k,t Observation The binary variables rit and cjt in PL can be relaxed to be nonnegative continuous PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds Linear Mixed Binary Program m T t t n T t t PL: max − lg rit − lg cjt 2T 2T 2T 2T Xi=1 Xt=1 jX=1 Xt=1 k s.t. xpq = 1 ∀ (i,j) ∈ B (kpqX)∈Iij T T k t rit = xpq , rit = 1 ∀i Xt=1 (kpqX)∈Ri Xt=0 T T k t cjt = xpq, cjt = 1 ∀ j Xt=1 (kpqX)∈Cj Xt=0 k xpq, rit, cjt ∈ {0, 1} ∀ i,j,p,q,k,t Observation The binary variables rit and cjt in PL can be relaxed to be nonnegative continuous PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds Exact Solution Methods Exact MIP solver (CPLEX 12.2) Exact MIP solver + New Branching Strategy Branch-and-Price (COIN-OR BCP 1.3.3) PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds Branching Strategy ! 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""))))))))))))))) "# "$ "% "&))))))))))))))) "' " # $ % " "( "* "+))))))))))))))) #! #" ##))))))))))))))) #$))))))))))))))) #% # & # $ #& #' #( #* #+))))))))))))))) $!))))))))))))))) $" $# $ % & $$ $% $& $' $( $* $+ %! % k = s (i, j) : 1 s ℓ, s Z; (i, j) f k , f k Mpq { → ≤ ≤ ∈ ∈ pq ∈ P} k Iij[s]= (k,p,q) : (i, j)= [s] { Mpq } PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds Branching Strategy Assume there exists (s, i, j) satisfying 0 < xk < 1 X pq (kpq)∈Iij [s] Branching k x = 0 (k,p,q) Iij[s] (Left Branch) pq ∀ ∈ k x = 0 (k,p,q) Iij Iij[s] (Right Branch) pq ∀ ∈ \ PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds Branching Strategy Assume there exists (s, i, j) satisfying 0 < xk < 1 X pq (kpq)∈Iij [s] Branching k x = 0 (k,p,q) Iij[s] (Left Branch) pq ∀ ∈ k x = 0 (k,p,q) Iij Iij[s] (Right Branch) pq ∀ ∈ \ PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds Computational Experiments: Setup Dual-core Intel Xeon E3110 3.00 GHz & 3 GB RAM Three polyomino families used: ℓ =4: tetromines ℓ =5: pentominoes ℓ =8: octominoes Since there are 2775 octominoes, only a subset of them is used PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds Computational Experiments: Setup Dual-core Intel Xeon E3110 3.00 GHz & 3 GB RAM Three polyomino families used: ℓ =4: tetromines ℓ =5: pentominoes ℓ =8: octominoes Since there are 2775 octominoes, only a subset of them is used PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds Exact Approaches: Tetromino Family Size Running Time (secs) # row # cols CPLEX CPLEX-BR BCP 12 12 1 2 74 12 18 1 1 13 12 24 10 10 18 16 16 6 6 27 16 19 17 17 21 16 20 8 7 46 16 25 33 33 110 20 20 138 106 21 20 23 1,071 153 33 20 24 38 37 60 20 30 818 2807 220 22 26 796 352 141 24 24 85 85 98 24 25 2,684 893 142 25 28 2,537 1,369 232 28 30 4226 299 30 30 329,839 275 32 32 935 40 40 830 40 60 7,979 40 80 54,270 40 100 374,919 PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds Exact Approaches: Pentomino Family Size Running Time (secs) # row # cols CPLEX CPLEX-BR BCP 10 10 1 1 8 10 13 3 4 13 13 15 68 17 45 15 15 15 16 15 15 17 14 15 43 15 19 18 19 17 17 20 85 96 98 20 20 64 73 225 20 24 1,670 996 389 20 30 5,221 2,361 620 23 25 294 293 206 25 25 112,694 5,103 365 25 29 3,521 173 30 30 12,938 331 40 40 6,200 40 60 14,133 40 80 22,000 40 100 103,725 50 100 540,070 60 80 442,864 PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds Exact Approaches: Octomino Family Size Running Time (secs) # row # cols CPLEX CPLEX-BR BCP 12 12 12 13 7 12 16 262 174 16 12 20 76 2,823 82 16 16 50 56 97 20 20 4,087 1,765 308 20 24 15,475 13,442 736 20 28 10,474 21,404 1,143 24 24 9,183 34,245 356 24 28 117,242 677 28 28 2,836 32 32 3,926 40 40 49,259 48 48 332,693 PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds Zoom-in Algorithm (ZiA) Strong Rep-tile Property (Golomb, 1970) A polyomino set is said to have the strong rep-tile property if P every member of can be tiled using at some common scale P P a b (we refer to it as zoom level) × PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds Zoom-in Algorithm (ZiA) Consider a tiling problem ( , ), = m n B P |B| × Assume has the strong rep-tile property at a b zoom level P × Let be a tiling of T0 B To obtain an am bn tiling using : replace each polyomino × P in with its tiling at a b zoom level T0 × An atm btn tiling using is available after t zoom-ins × P PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds Meta-rectangle Tiling Algorithm (MrTA) Motivated by the meta-rectangle idea developed in the context of the classical cutting stock problem [Wang, 1983] = r = (x ,y ), . . . , rk = (xk,yk) - a set of rectangles R { 1 1 1 } for which a tiling using exist P Any meta-rectangle obtained by juxtaposing available rectangles and meta-rectangles horizontally and vertically also has a tiling using P PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds Meta-rectangle Tiling Algorithm (MrTA) Juxtapose two rectangles horizontally and then repeat them vertically until the least common multiplier of their vertical dimension is reached , - ! .(+(,'(- /(+(-'(!(0123245 6(+(.'(/ "#$%7'(&*(+(8 "#$%&'()*(+(& "#$%8'(&*(+(8 PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds 1 160 1 160 All meta-rectangles created using only {(4, 10), (7, 8)} PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds Improvement Heuristic: Retiling A tiling for problem ( , ) T B P A window w of size (2d + 1) (2d + 1) located at (r, c) × ∈B All polyominoes in and crossing the boundary of w are fixed T Define a new entropy maximizing problem to tile interior of w The new tiling problem should be small enough to be efficiently solvable Retile moving along rows and columns of with step sizes ∆r B and ∆c PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds Improvement Heuristic: Retiling A tiling for problem ( , ) T B P A window w of size (2d + 1) (2d + 1) located at (r, c) × ∈B All polyominoes in and crossing the boundary of w are fixed T Define a new entropy maximizing problem to tile interior of w The new tiling problem should be small enough to be efficiently solvable Retile moving along rows and columns of with step sizes ∆r B and ∆c PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds 120 × 160 meta-rectangle. PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds 120 × 160 octomino tiling of the meta-rectangle PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds The tiling randomized through to its middle PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds Centers of gravity PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds Paste Side-by-side Algorithm (PSA) A simplified version of MrTA MrTA with only one rectangle and rotations are not allowed Consider a tiling of a board , = m n T0 B |B| × Entropy of : T0 m r r n c c E = j lg j j lg j 0 − X 2T 2T − X 2T 2T j=1 j=1 Let be an am bn tiling obtained by pasting a times T1 × T0 vertically and b times horizontally Entropy of : T1 E1 = E0 + lg √a b PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds Paste Side-by-side Algorithm (PSA) A simplified version of MrTA MrTA with only one rectangle and rotations are not allowed Consider a tiling of a board , = m n T0 B |B| × Entropy of : T0 m r r n c c E = j lg j j lg j 0 − X 2T 2T − X 2T 2T j=1 j=1 Let be an am bn tiling obtained by pasting a times T1 × T0 vertically and b times horizontally Entropy of : T1 E1 = E0 + lg √a b PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds PSA: Approximation Bounds Proposition If the entropy of the initial tiling is at least ( 1 + ǫ) lg(m + n), T0 2 ǫ [0, 1 ], then PSA has an approximation guarantee of ∈ 2 1 lg(abm + abn) lg(m + n) 1 + ǫ 2 lg(am + bn) lg(am + bn) ≥ 2 Corollary If a = b and the initial tiling is ǫ-optimal with respect to the T0 theoretical upper bound, then the solution obtained by PSA is also ǫ-optimal PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds PSA: Approximation Bounds Proposition If the entropy of the initial tiling is at least ( 1 + ǫ) lg(m + n), T0 2 ǫ [0, 1 ], then PSA has an approximation guarantee of ∈ 2 1 lg(abm + abn) lg(m + n) 1 + ǫ 2 lg(am + bn) lg(am + bn) ≥ 2 Corollary If a = b and the initial tiling is ǫ-optimal with respect to the T0 theoretical upper bound, then the solution obtained by PSA is also ǫ-optimal PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds Summary Future work: compare radiation patterns for various types of generated irregular polyomino tilings Reference: S. Karademir, O. Prokopyev, “Irregular Polyomino Tiling via Integer Programming,” Technical Report, 2012. PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds Two-Stage Stochastic Integer Programs We consider the following class of two-stage stochastic programs with integer variables in both stages (2SSIP): T (P1) : max c x + IEξQ(x,ξ(ω)) subject to Ax b ≤ n1 x ZZ+ where Q(x,ξ(ω)) is defined as: ∈ Q(x,ξ(ω)) = max dT y subject to Wy h(ω) Tx ≤ − y ZZn2 ∈ + n1 n2 m1 m1×n1 m2×n1 c ZZ+ , d ZZ+ , b ZZ+ , A ZZ+ , T ZZ+ , ∈ m ∈n ∈ ∈ ∈ W ZZ 2× 2 ∈ + Stochasticity encountered solely in right-hand sides h(ω) ZZm2 ∈ + Random variable ω follows discrete distribution with finite support Ω PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds Two-Stage Stochastic Integer Programs We consider the following class of two-stage stochastic programs with integer variables in both stages (2SSIP): T (P1) : max c x + IEξQ(x,ξ(ω)) subject to Ax b ≤ n1 x ZZ+ where Q(x,ξ(ω)) is defined as: ∈ Q(x,ξ(ω)) = max dT y subject to Wy h(ω) Tx ≤ − y ZZn2 ∈ + n1 n2 m1 m1×n1 m2×n1 c ZZ+ , d ZZ+ , b ZZ+ , A ZZ+ , T ZZ+ , ∈ m ∈n ∈ ∈ ∈ W ZZ 2× 2 ∈ + Stochasticity encountered solely in right-hand sides h(ω) ZZm2 ∈ + Random variable ω follows discrete distribution with finite support Ω PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds Two-Stage Stochastic Integer Programs We consider the following class of two-stage stochastic programs with integer variables in both stages (2SSIP): T (P1) : max c x + IEξQ(x,ξ(ω)) subject to Ax b ≤ n1 x ZZ+ where Q(x,ξ(ω)) is defined as: ∈ Q(x,ξ(ω)) = max dT y subject to Wy h(ω) Tx ≤ − y ZZn2 ∈ + n1 n2 m1 m1×n1 m2×n1 c ZZ+ , d ZZ+ , b ZZ+ , A ZZ+ , T ZZ+ , ∈ m ∈n ∈ ∈ ∈ W ZZ 2× 2 ∈ + Stochasticity encountered solely in right-hand sides h(ω) ZZm2 ∈ + Random variable ω follows discrete distribution with finite support Ω PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds Value Function Definitions for First Stage Without loss of generality, any first-stage constraints Ax b can be ≤ incorporated into T , W and h(ω), thus we let m = m1 + m2 β ZZm, define the first-stage value function as: ∀ 1 ∈ ψ(β ) = max cT x x S (β ) , where 1 | ∈ 1 1 S (β )= x ZZn1 Tx β 1 1 ∈ + | ≤ 1 Define ui = min hi(ω), i =1,...,m ω∈Ω ∀ B1 m 0 u m B1 Define = i=1[ i, i] ZZ ; then is the set of vectors m Q T β1 ZZ+ such that x S1(β1) with β1 h(ω) for all ω Ω ∈ ∃ ∈ ≤ ∈ PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds Value Function Definitions for First Stage Without loss of generality, any first-stage constraints Ax b can be ≤ incorporated into T , W and h(ω), thus we let m = m1 + m2 β ZZm, define the first-stage value function as: ∀ 1 ∈ ψ(β ) = max cT x x S (β ) , where 1 | ∈ 1 1 S (β )= x ZZn1 Tx β 1 1 ∈ + | ≤ 1 Define ui = min hi(ω), i =1,...,m ω∈Ω ∀ B1 m 0 u m B1 Define = i=1[ i, i] ZZ ; then is the set of vectors m Q T β1 ZZ+ such that x S1(β1) with β1 h(ω) for all ω Ω ∈ ∃ ∈ ≤ ∈ PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds Value Function Definitions for First Stage Without loss of generality, any first-stage constraints Ax b can be ≤ incorporated into T , W and h(ω), thus we let m = m1 + m2 β ZZm, define the first-stage value function as: ∀ 1 ∈ ψ(β ) = max cT x x S (β ) , where 1 | ∈ 1 1 S (β )= x ZZn1 Tx β 1 1 ∈ + | ≤ 1 Define ui = min hi(ω), i =1,...,m ω∈Ω ∀ B1 m 0 u m B1 Define = i=1[ i, i] ZZ ; then is the set of vectors m Q T β1 ZZ+ such that x S1(β1) with β1 h(ω) for all ω Ω ∈ ∃ ∈ ≤ ∈ PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds Value Function Definitions for Second Stage β ZZm, define the second-stage value function as: ∀ 2 ∈ φ(β ) = max dT y y S (β ) , where 2 | ∈ 2 2 S (β )= y ZZn2 Wy β 2 2 ∈ + | ≤ 2 2 2 B 1 B Define = β1∈B ω∈Ω h(ω) β1 , i.e., is the set of vectors m ∪ ∪ { 1 − } β2 ZZ such that β1 B and ω Ω, β2 = h(ω) β1 ∈ ∃ ∈ ∈ − PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds Value Function Reformulation of (P1) (P1) can then be reformulated as: (P2): max ψ(β1)+ IEξφ(h(ω) β1) 1 β1∈B { − } Note that variables β1 are now m-dimensional right-hand sides Theorem (Ahmed et al. 2004; Kong et al. 2006) ∗ T Let β1 be an optimal solution to (P2). Then xˆ arg max c x ∗ n1 ∈ { | Tx β1 , x ZZ+ is an optimal solution to (P1). Furthermore, the optimal≤ objective∈ values} of the two problems are equal. PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds Value Function Reformulation of (P1) (P1) can then be reformulated as: (P2): max ψ(β1)+ IEξφ(h(ω) β1) 1 β1∈B { − } Note that variables β1 are now m-dimensional right-hand sides Theorem (Ahmed et al. 2004; Kong et al. 2006) ∗ T Let β1 be an optimal solution to (P2). Then xˆ arg max c x ∗ n1 ∈ { | Tx β1 , x ZZ+ is an optimal solution to (P1). Furthermore, the optimal≤ objective∈ values} of the two problems are equal. PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds Value Function Reformulation of (P1) (P1) can then be reformulated as: (P2): max ψ(β1)+ IEξφ(h(ω) β1) 1 β1∈B { − } Note that variables β1 are now m-dimensional right-hand sides Theorem (Ahmed et al. 2004; Kong et al. 2006) ∗ T Let β1 be an optimal solution to (P2). Then xˆ arg max c x ∗ n1 ∈ { | Tx β1 , x ZZ+ is an optimal solution to (P1). Furthermore, the optimal≤ objective∈ values} of the two problems are equal. PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds Use of (P2) Reformulation in the Literature Ahmed, Tawarmalani, Sahinidis: Math. Programming ’04 [W /d/h] Mixed-integer first-stage and pure integer second-stage variables Exploit reformulation to handle discontinuities of value function Global branch and bound; lower corner cuts bound IEξQ(x, ξ(ω)) Kong, Schaefer, Hunsaker: Math. Programming ’06 [h] Pure integer variables in both stages For all β in both stages, find and store complete value function ∗ Use (i) global branch and bound, (ii) level-set approaches to find β1 Ozaltın,¨ Prokopyev, Schaefer: Math. Programming ’10 [h] Pure integer variables in both stages ψ( ) and φ( ) include quadratic terms · · For all β in both stages, find and store complete value function ∗ Use (i) global branch and bound, (ii) level-set approaches to find β1 PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds Use of (P2) Reformulation in the Literature Ahmed, Tawarmalani, Sahinidis: Math. Programming ’04 [W /d/h] Mixed-integer first-stage and pure integer second-stage variables Exploit reformulation to handle discontinuities of value function Global branch and bound; lower corner cuts bound IEξQ(x, ξ(ω)) Kong, Schaefer, Hunsaker: Math. Programming ’06 [h] Pure integer variables in both stages For all β in both stages, find and store complete value function ∗ Use (i) global branch and bound, (ii) level-set approaches to find β1 Ozaltın,¨ Prokopyev, Schaefer: Math. Programming ’10 [h] Pure integer variables in both stages ψ( ) and φ( ) include quadratic terms · · For all β in both stages, find and store complete value function ∗ Use (i) global branch and bound, (ii) level-set approaches to find β1 PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds Use of (P2) Reformulation in the Literature Ahmed, Tawarmalani, Sahinidis: Math. Programming ’04 [W /d/h] Mixed-integer first-stage and pure integer second-stage variables Exploit reformulation to handle discontinuities of value function Global branch and bound; lower corner cuts bound IEξQ(x, ξ(ω)) Kong, Schaefer, Hunsaker: Math. Programming ’06 [h] Pure integer variables in both stages For all β in both stages, find and store complete value function ∗ Use (i) global branch and bound, (ii) level-set approaches to find β1 Ozaltın,¨ Prokopyev, Schaefer: Math. Programming ’10 [h] Pure integer variables in both stages ψ( ) and φ( ) include quadratic terms · · For all β in both stages, find and store complete value function ∗ Use (i) global branch and bound, (ii) level-set approaches to find β1 PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds Advantages and Limitations of Reformulation Advantages: relatively insensitive to n , n , and Ω 1 2 | | Both Kong et al. and Ozaltın¨ et al. can solve instances with n1 = 1000, n2 = 500, and Ω = 279, 936 | | Limitations: somewhat sensitive to both m and hi(ω) values | | Both Kong et al. and Ozaltın¨ et al. report handling up to 7 rows Memory limitations unfortunately prevent solving larger instances We attempt to address the limitation on m through: 1 A level-set approach (Trapp, Prokopyev, Schaefer: Tech. Report) up to 50-100 rows (depending on the constraint matrix structure) 2 Constraint aggregation approach (Trapp, Prokopyev) working paper PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds Advantages and Limitations of Reformulation Advantages: relatively insensitive to n , n , and Ω 1 2 | | Both Kong et al. and Ozaltın¨ et al. can solve instances with n1 = 1000, n2 = 500, and Ω = 279, 936 | | Limitations: somewhat sensitive to both m and hi(ω) values | | Both Kong et al. and Ozaltın¨ et al. report handling up to 7 rows Memory limitations unfortunately prevent solving larger instances We attempt to address the limitation on m through: 1 A level-set approach (Trapp, Prokopyev, Schaefer: Tech. Report) up to 50-100 rows (depending on the constraint matrix structure) 2 Constraint aggregation approach (Trapp, Prokopyev) working paper PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds Advantages and Limitations of Reformulation Advantages: relatively insensitive to n , n , and Ω 1 2 | | Both Kong et al. and Ozaltın¨ et al. can solve instances with n1 = 1000, n2 = 500, and Ω = 279, 936 | | Limitations: somewhat sensitive to both m and hi(ω) values | | Both Kong et al. and Ozaltın¨ et al. report handling up to 7 rows Memory limitations unfortunately prevent solving larger instances We attempt to address the limitation on m through: 1 A level-set approach (Trapp, Prokopyev, Schaefer: Tech. Report) up to 50-100 rows (depending on the constraint matrix structure) 2 Constraint aggregation approach (Trapp, Prokopyev) working paper PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds Advantages and Limitations of Reformulation Advantages: relatively insensitive to n , n , and Ω 1 2 | | Both Kong et al. and Ozaltın¨ et al. can solve instances with n1 = 1000, n2 = 500, and Ω = 279, 936 | | Limitations: somewhat sensitive to both m and hi(ω) values | | Both Kong et al. and Ozaltın¨ et al. report handling up to 7 rows Memory limitations unfortunately prevent solving larger instances We attempt to address the limitation on m through: 1 A level-set approach (Trapp, Prokopyev, Schaefer: Tech. Report) up to 50-100 rows (depending on the constraint matrix structure) 2 Constraint aggregation approach (Trapp, Prokopyev) working paper PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds Advantages and Limitations of Reformulation Advantages: relatively insensitive to n , n , and Ω 1 2 | | Both Kong et al. and Ozaltın¨ et al. can solve instances with n1 = 1000, n2 = 500, and Ω = 279, 936 | | Limitations: somewhat sensitive to both m and hi(ω) values | | Both Kong et al. and Ozaltın¨ et al. report handling up to 7 rows Memory limitations unfortunately prevent solving larger instances We attempt to address the limitation on m through: 1 A level-set approach (Trapp, Prokopyev, Schaefer: Tech. Report) up to 50-100 rows (depending on the constraint matrix structure) 2 Constraint aggregation approach (Trapp, Prokopyev) working paper PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds Advantages and Limitations of Reformulation Advantages: relatively insensitive to n , n , and Ω 1 2 | | Both Kong et al. and Ozaltın¨ et al. can solve instances with n1 = 1000, n2 = 500, and Ω = 279, 936 | | Limitations: somewhat sensitive to both m and hi(ω) values | | Both Kong et al. and Ozaltın¨ et al. report handling up to 7 rows Memory limitations unfortunately prevent solving larger instances We attempt to address the limitation on m through: 1 A level-set approach (Trapp, Prokopyev, Schaefer: Tech. Report) up to 50-100 rows (depending on the constraint matrix structure) 2 Constraint aggregation approach (Trapp, Prokopyev) working paper PITT