Searching for the Minimal B

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Searching for the Minimal Bézout Number Lin
Zhenjiang, Allenzjlin_at_cse.cuhk.edu.hk Dept. of
CSE, CUHK 3-Oct-2005
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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

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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

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Polynomial system problem
Mission Find out all solutions of P(X).
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Application very common in many engineering
fields

• formula construction,
• geometric intersection problems,
• computation of equilibrium states,
• etc.

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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

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Homotopy method

• Homotopy Equation

Construct Q(x) that satisfy the conditions

1. The solutions of Q(x) 0 are either known or
   easy to known
2. When 0t 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.

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Figure 1 Illustration of Homotopy method
Mission Construct Q(X) with minimal number of
solutions.
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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

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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.
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More detail ---

• Divide X (x1, x2, , xn) into m groups
• X (X (1), X(2), ,X(m)),
• then we get

1. the degree matrix D ( dij ), where dij is the
   degree of X (j) in Pi(X (1), X(2), ,X(m))
2. and partition vector K (kj)T, where kj is the
   number of variables that X (j) contains.

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Example 1 ( n 3)
If X (x1, x2, x3) is divide into 2 groups
X ( x1, x2 , x3 ), or ( 1, 2 ,
3 ),then we have
and
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• The formula for Bézout number B(D,K) is

where
and Per(D) is the permanent of matrix D.
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and
In Example 1,

• So, the Bézout number is

where
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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

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Problems

• Searching the optimal one in all possible
  partition strategies
• Model How many ways to put n balls into m
  (1mn ) boxes? The result is called the Bell
  number B(n), which has the following estimation
• (n / 2) (n / 2) lt B(n) lt n!

• Computing Bézout number (or permanent)
• The best-known algorithm is Rysers, O(n2n).

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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

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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.

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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.
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Evaluation function

• Bézout number, right?

But how can we calculate it ?
Thats our next problem.
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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

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Monte Carlo method for Bézout number calculating

• Bézout number and permanent

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• Permanent
• where A is an nn matrix, and Sn is the set
  composed of all permutations of number 1, 2,,n.
• Example 2.

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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

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• We can see in the definition of permanent
• any permutation of 1, 2,,n, denoted by?,
  corresponds to one product term g(?)
• therere totally n! product terms.
• Let Sn be the sample space ?. We have
• Per(A) ? ? ? n! (6.3)
• where? E(g(?)) is the expectation of g(?).

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MC (Monte Carlo) Method

• where
• is the approximation of by sampling
  uniformly from sample space ?.

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Disadvantage of MC method

• Too many zero-value product terms when matrix A
  is sparse, i.e., for an nn matrix with sparsity
  p,
• pn pn?0, n ?8
• pn is the possibility of sampling a non-zero
  sample.
• Applying simple Monte Carlo approach to our
  problem is not very helpful.

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MC(?) algorithm
Let to be the sample space, then we
have where Advantage ? ltlt ? Question
How to get and ? ?
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How to get ? ?
Let IA is a matrix that has the same structure as
A except that the non-zero entries is
1. Obviously, we have ? Per(IA) Thus we can
calculate ? with 0-1 permanent algorithms.
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How to get or ?
The equivalent question is How can we choose a
non-zero product term uniformly?
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Expand a permanent on the first column For any
matrix A(aij) nn, 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?
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Example,
then,
Divide product terms into 3 groups!
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? ?
? 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.
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Layered MC(?) algorithm

• Basic Idea - Importance sampling
• Divide the sample space ? into several
  sub-spaces, in which the sample values are closer.

Ai Keep the i largest entries of A, and others
set to zero
? is divided into n2 sub-spaces according to the
value of product terms.
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Layered MC(?) 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.

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Numerical results
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1. Conclusion

• Polynomial systems
• Homotopy method Bézout theory
• Searching for minimal Bézout number
• Tabu search
• Computing Bézout number
• Monte Carlo method