Irregular Polyomino Tiling via Integer Programming Oleg Prokopyev Serdar Karademir Department of Industrial Engineering University of Pittsburgh [email protected] AFOSR Optimization and Discrete Mathematics Program Review April 18, 2012 PITT Motivation Optimization Model Exact Methods Heuristics Approximation Bounds Polyominoes: , , ... F1 F2 !"#"$%#" &"$%#" '("$%#")* /)#-"$%#")* (%+,- . ' < / 6 = ')-("$%#")* 2 . 7 ; 9 : *0)1 *345() 8 . 2 ' A generalization of the domino Created by connecting unit squares along edges 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<n; i, j Z ≤ ∈ } & '&(!"#) 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<n; i, j Z ≤ ∈ } & '&(!"#) 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.