A Pigeon-Hole Based Encoding of Cardinality Constraints 1

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1 A Pigeon-Hole Based Encoding of Cardinality Constraints 1 Saïd Jabbour Lakhdar Saïs Yakoub Salhi January 7, 2014 CRIL - CNRS, Université Lille Nord de France, Lens, France 1 Funded by ANR DEFIS 2009 Program, DAG Project Declarative Approaches for Enumerating Interesting Patterns 1 / 13

2 Motivation CP/SAT based data mining (Frequency constraint) Cardinality constraints appears in many other application domains Cross-fertilization between 0/1 linear programming and SAT Goal : Find the most efficient and compact CNF encoding (Efficient) Maintain generalized arc consistency via unit propagation (Compact) Encoding of smallest size? (w.r.t. number of variables and clauses) 2 / 13

3 CNF encodings of the cardinality constraint n x i λ, x i {0, 1} i=1 Several polynomial CNF encodings have been proposed: Joost P. Warners [1996] [Wrong formulation in page 12] BDD encoding [Bailleux at al. 2003]... Cardinality networks [Asín et al. 2011] 3 / 13

4 Joost P. Warners [1996] - page 12 4 / 13

5 Polynomial Encoding of n i=1 x i λ to CNF λ ( p ki x i ), i = 1,..., n (1) k=1 1 k<k λ n p ki, k = 1,..., λ (2) i=1 ( p ki p k i), i = 1,..., n (3) (2) and (3) encode the Pigeon Hole problem PHP λ n p ki expresses that pigeon k is in hole i x i is true if the hole i contains one of the pigeons k for k = 1,..., λ Complexity number of additional variables: λ n number of clauses: O(n λ 2 ). 5 / 13

6 Symmetry Breaking on P λ n Let Sym(Pn λ ) be the set of symmetries of Pn λ : σ(i, j) = (p i1, p j1 ), (p i2, p j2 ),..., (p in, p jn ) (4) 1 i<j λ p 11 [p 1λ p 1n ] [p 2(λ 1) p 2(n 1) ] (5) [p λ1 p λ(n λ+1) ] p λn 6 / 13

7 Symmetry Breaking on P λ n (cont.) Let σ(i, j) = (p i1, p j1 ), (p i2, p j2 ),..., (p in, p jn ), sbp σ(i,j) = (p i1 p j1 ) (p i1 = p j1 ) (p i2 p j2 )... Property (p i1 = p j1 )... (p i(n 1) = p j(n 1) ) (p in p jn ) P λ n is satisfiable iff P λ n sbp(sym(p λ n )) is satisfiable 1 i < j λ Instead of adding SBP to the formula, we apply resolution between clauses from P λ n and sbp(sym(p λ n )) 7 / 13

8 Symmetry Breaking on P λ n (cont.) Eliminating the upper-left corner triangle P λ n sbp σ(1,2) = p 11 : σ = (p 11, p 21 ) σ(1, 2) sbp σ = (p 12 p 22 ) = (1)( p 11 p 21 ) c = ( p 11 p 21 ) P λ n η[p 21, (1), c] = p 11. P b n sbp(sym(p b n )) = p 21 p 31,..., p (λ 1)1 Use similar reasoning All the binary clauses from (3) involving p 11, p 21,..., p (λ 1)1 are eliminated... 8 / 13

9 Symmetry Breaking on P λ n (cont.) Eliminating the lower-right corner triangle P λ n sbp(sym(pλ n )) = p 2n p 3n,..., p λn P λ n sbp(sym(p λ n )) = p 2n : η[p 1n, (p 1λ,..., p 1(n 1) p 1n), ( p 1n p 2n)] = r 1 = (p 1λ,..., p 1(n 1) p 2n). η[p 2n, r 1, (p 2(λ 1),..., p 2(n 1) p 2n)] = r 2 = (p 1λ,..., p 1(n 2) p 1(n 1) p 2(λ 1),..., p 2(n 1)). To eliminate the first n 1 literals from r 2, we exploit sbp σ(1,i) with 2 i λ. Let s 1 = (p 2(λ 1) p 1λ p 2λ..., p 1(n 2) p 2(n 2) p 1(n 1) p 2(n 1) ) sbp σ(1,2). η[p 1(n 1), r 2, s 1] = r 3 = (p 1λ,..., p 1(n 2) p 2(λ 1),..., p 2(n 1) ). Now, p 1(n 2) can be eliminated from r 3. Let s 2 = (p 2(λ 1) p 1λ p 2λ..., p 1(n 3) p 2(n 3) p 1(n 2) p 2(n 1) ) sbp σ(1,2). We obtain η[p 1(n 2), r 3, s 2] = r 4 = (p 1λ,..., p 1(n 3) p 2(λ 1),..., p 2(n 1) ). Similarly, we eliminate {p 1λ,..., p 1(n 3) } from r / 13

10 php λ n encoding of the cardinality constraint p (λ k+1)(i+k 1) x (i+k 1), 1 i n λ + 1, 1 k λ (6) n λ+1 i=1 p (λ k+1)(i+k 1), 1 k λ (7) p (λ k+1)k p (λ k+1)(i+k) p (λ k)(i+k+1), 0 i n λ 1, 1 k λ 1 (8) 10 / 13

11 Soundness and Unit Propagation Property If ρ is a model of php λ n then ρ is a model of n i=1 x i λ. Property If ρ is a model of n i=1 x i λ, then there exists a model ρ of php λ n such that for all i {1,..., n}, ρ(x i ) = ρ (x i ). Property (Unit propagation) Let ρ be a model of php λ n assigning 0 to the elements of a set S = {x i1,..., x in λ } of n λ propositional variables included in X = {x 1,..., x n }. Unit propagation is sufficient to deduce that for all variable x X \S, ρ(x) = / 13

12 Theoretical Comparison with other encodings Encoding #Clauses # Variables Decided Sequential unary counter [Sinz05] O(λ n) O(λ n) UP Parallel binary counter [Sinz05] 7n 3log(n) 6 2n 2 Search Totalizer [Bailleux03] O(n 2 ) O(n log 2 (n)) UP Buttner & Rintanen [Buttner05] O(λ 2 n) O(n log 2 (n)) UP Sorting Network [EenS06] O(n log2 2(n)) O(n log 2 2 (n)) UP Cardinality Network [AsinNOR11] O(n log2 2(λ)) O(n log 2 2 (λ)) UP Warners [Warners96] 8n 2n Search php λ n O(λ (n λ)) O(λ (n λ)) UP Table : Comparison of CNF encodings of n i=1 x i λ 12 / 13

13 Conclusion Conclusion & future works Pigeon Hole Based formulation of the cardinality constraint Competitive with most of the previous encoding Erratum to Joost P. Warners formulation [1996 paper] A nice methodology: reduction of the encoding modulo symmetry, redundant constraints, resolution... Future works CNF encoding of n i=1 α ix i λ for both x i {0, 1} and x i {1..., n i } (done - Best encoding) Use of the same methodology to encode global constraints to CNF (e.g. alldifferent constraint) 13 / 13