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Searching for the Minimal Bézout Number Lin Zhenjiang, Allen zjlin@cse.cuhk.edu.hk Dept. of CSE, CUHK 3-Oct-2005 1. Polynomial system problem 2. Homotopy method 3. Bézout theory and minimal Bézout number 4. Problems 5. Tabu search method for minimal Bézout number searching 6. Monte Carlo method for Bézout number calculating 7. Conclusion 1. Polynomial system problem 2. Homotopy method 3. Bézout theory and minimal Bézout number 4. Problems 5. Tabu search method for minimal Bézout number searching 6. Monte Carlo method for Bézout number calculating 7. Conclusion 1. Polynomial system problem p1 ( x1 ,, xn ) 0 p ( x ,, x ) 0 2 1 n P( X ) : , (1.1) pn ( x1 ,, xn ) 0 where X {x1 ,, xn } Mission: Find out all solutions of P(X). Application: very common in many engineering fields formula construction, geometric intersection problems, computation of equilibrium states, etc. 1. Polynomial system problem 2. Homotopy method 3. Bézout theory and minimal Bézout number 4. Problems 5. Tabu search method for minimal Bézout number searching 6. Monte Carlo method for Bézout number calculating 7. Conclusion 2. Homotopy method Homotopy Equation: H ( x, t ) (1 t )Q( x) tP( x) Construct Q(x) that satisfy the conditions: 1. The solutions of Q(x) = 0 are either known or easy to known; 2. When 0≤t ≤1, the solutions of H(x,t) is consist of finite number of curves with parameter t; 3. Each solution of H(x, 1)= P(x) = 0 can be obtained by tracking curves starting from t = 0. t H(x,1) = P(x)= 0 t=1 1111 1111 H(x,t) = 0 = t=0 H(x,0) = Q(x)= 0 Figure 1: Illustration of Homotopy method Mission: Construct Q(X) with minimal number of solutions. 1. Polynomial system problem 2. Homotopy method 3. Bézout theory and minimal Bézout number 4. Problems 5. Tabu search method for minimal Bézout number searching 6. Monte Carlo method for Bézout number calculating 7. Conclusion 3. Minimal Bézout number For a polynomial system P(X) = 0, where P = (p1, p2, …, pn), X = (x1, x2, …, xn), Bézout theory By dividing the n variables x1, x2, …, xn into several groups (called a partition strategy), we can get the corresponding Q(X) and an upper bound of its solution number - Bézout number. Mission: Find out the partition strategy which corresponds to the minimal Bézout number. More detail --- Divide X = (x1, x2, …, xn) into m groups: X = (X (1), X(2), …,X(m)), then we get a) the degree matrix D = ( dij ), where dij is the degree of X (j) in Pi(X (1), X(2), …,X(m)); b) and partition vector K = (kj)T, where kj is the number of variables that X (j) contains. x1 x2 2 x1 x3 x2 0 Example 1: 2 P( X ) : x1 x3 x2 x3 0 2 ( n = 3) x1 x2 x3 0 2 If X = (x1, x2, x3) is divide into 2 groups: X = ( { x1, x2 }, { x3 } ), or ( { 1, 2 }, { 3 } ), then we have 3 1 K 2, 1 T D 2 1 and 1 2 The formula for Bézout number B(D,K) is Per D* 1 B ( D, K ) k1!k 2 ! k m ! where d d d d d d 11 11 12 12 1m 1m D* n1 d n dn d nm d 1 d n 2 2 d nm k k1 2 km n n and Per(D*) is the permanent of matrix D*. 3 1 D 2 1 and K 2, 1 T In Example 1, 1 2 So, the Bézout number is B ( D, K ) 1 2 ! 1! Per D 34 * 3 3 1 where D 2 2 1 * 1 1 2 1. Polynomial system problem 2. Homotopy method 3. Bézout theory and minimal Bézout number 4. Problems 5. Tabu search method for minimal Bézout number searching 6. Monte Carlo method for Bézout number calculating 7. Conclusion 4. Problems Searching the optimal one in all possible partition strategies Model: How many ways to put n balls into m (1≤m≤n ) boxes? The result is called the Bell number B(n), which has the following estimation (n / 2) (n / 2) < B(n) < n! Computing Bézout number (or permanent) The best-known algorithm is Ryser’s, O(n2n). 1. Polynomial system problem 2. Homotopy method 3. Bézout theory and minimal Bézout number 4. Challenges 5. Tabu search method for minimal Bézout number searching 6. Monte Carlo method for Bézout number calculating 7. Conclusion 5. Tabu search method for minimal Bézout number searching Main idea: Construct neighbor relationship between partition strategies (or partitions), and apply Tabu (Taboo) search method to search the optimal partition. Two kinds of neighbor relationship split: { 1, 3, 6 } { 5 } { 2, 4 } ↙↘ { 1, 3 } { 6 } { 5 } { 2, 4 } merge: { 1, 3, 6 } { 5 } { 2, 4 } ↘↙ { 1, 3 , 6 } { 5, 2, 4 } A partition has O(n2) neighbors. Evaluation function Bézout number, right? But how can we calculate it ? That’s our next problem. 1. Polynomial system problem 2. Homotopy method 3. Bézout theory and minimal Bézout number 4. Challenges 5. Tabu search method for minimal Bézout number searching 6. Monte Carlo method for Bézout number calculating 7. Conclusion 6. Monte Carlo method for Bézout number calculating Bézout number and permanent 1 B ( D, K ) * Per ( D ) (6.1) k1!k 2! k m ! Permanent a n Per( A) i i , (i ) (6.2) S n where A is an n×n matrix, and Sn is the set composed of all permutations of number 1, 2,…,n. Example 2. 2 3 4 5 , Per( A) 2 5 4 3 22 A More about the permanent The computation of permanents has been studied fairly extensively in algebraic complexity theory. The complexity of the best-known algorithms grows as the exponent of the matrix size. Application – Counting problems The number of perfect matching -- 0-1 permanent The number of Latin squares -- general permanent We can see in the definition of permanent a. any permutation of 1, 2,…,n, denoted byω, corresponds to one product term g(ω); b. there’re totally n! product terms. Let Sn be the sample space Ω. We have Per(A) = θ· |Ω| = θ· n! (6.3) whereθ= E(g(ω)) is the expectation of g(ω). MC (Monte Carlo) Method Per( A) n! (6.4) where N 1 N g ( ) i 1 (6.5) is the approximation of by sampling uniformly from sample space Ω. Disadvantage of MC method Too many zero-value product terms when matrix A is sparse, i.e., for an n×n matrix with sparsity p, pn ≤ pn→0, n →∞ pn is the possibility of sampling a non-zero sample. Applying simple Monte Carlo approach to our problem is not very helpful. MC(Ω+) algorithm Let g () 0, to be the sample space, then we have Per( A) (6.6) where N 1 N g ( ) i 1 (6.7 ) Advantage |Ω+| << |Ω| + Question How to get and |Ω | ? How to get |Ω+ | ? Let IA is a matrix that has the same structure as A except that the non-zero entries is 1. Obviously, we have |Ω+ | = Per(IA) Thus we can calculate |Ω+ | with 0-1 permanent algorithms. How to get or g ( ) ? The equivalent question is How can we choose a non-zero product term uniformly? How to get or g ( ) ? Expand a permanent on the first column For any matrix A=(aij) n×n, we have Per(A) = ∏ ai1Per(Ac(ai1)), ∨a∈A where Ac(ai1) is the complementary sub-matrix of A about ai1. Remember Laplace expansion on a determinant? How to get or g ( ) ? Example, 3 6 1 A 2 4 0 0 1 5 then, 4 0 6 1 6 1 1 5 2 Per 1 5 0 Per 4 0 Per( A) 3 Per Divide product terms into 3 groups! How to get or g ( ) ? 4 0 6 1 6 1 1 5 2 Per 1 5 0 Per 4 0 Per( A) 3 Per ↓ ↓ ↓ 1 product term 2 terms 0 term Choose “3” (group 1) with the probability 1/3, “2” (group 2) with 2/3, and “0” (group 3) with 0. By iterating this procedure, we can finally sample uniformly a none-zero product term. Layered MC(Ω+) algorithm Basic Idea - Importance sampling Divide the sample space Ω+ into several sub-spaces, in which the sample values are closer. n2 n2 Per ( A) ai Per ( A (ai )) g ( ) i c i 1 i 1 i Ai – Keep the i largest entries of A, and others set to zero; Product terms in sub-space i ( A) are likely larger than those in sub-space i1 ( A) . Ω+ is divided into n2 sub-spaces according to the value of product terms. Layered MC(Ω+) algorithm How to assign sampling number to sub-spaces? Based on the dimension of sub-spaces; the sums of product terms that have already been estimated in sub-spaces; Lead to two algorithms: M1 and M2. Numerical results 7. Conclusion Polynomial systems Homotopy method & Bézout theory Searching for minimal Bézout number Tabu search Computing Bézout number Monte Carlo method

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