The Project-And-Lift Algorithm for the Computation of Toric Gr¨ obner Bases An Implementation in Mathematica Stephan Ritscher Department of Computer Science TU M¨ unchen 13th of April 2010 Stephan Ritscher: Computation of Toric Gr¨ obner Bases 1/ 29
Outline 1 Motivation 2 Prerequisites 3 Problem Statement 4 Overview of Algorithms Common Parts Saturation Algorithm Shadow Algorithm Project-and-Lift Algorithm Geometric Buchberger Algorithm 5 Experimental Results 6 Conclusion Stephan Ritscher: Computation of Toric Gr¨ obner Bases 2/ 29
Motivation Integer Linear Programming Standard Form min { cy : Ay = b , y ∈ N n } Selected Applications • Travelling Salesman • Knapsack, Bin Packing • Schedule Optimization • Frequency Planning for Mobile Phone Networks • Capacity Planning for Telecommunication Networks Stephan Ritscher: Computation of Toric Gr¨ obner Bases 3/ 29
Motivation Applying Gr¨ obner-Bases Definition (Test Set) T is a test set for min { cy : Ay = b , y ∈ N n } if 1 all y ∈ T have negative cost ( cy < 0), 2 all y ∈ T solve Ay = 0 and 3 for each non-optimal solution y s ∈ N n of Ay s = b there is a y ∈ T s.t. y s + y ∈ N n . Stephan Ritscher: Computation of Toric Gr¨ obner Bases 4/ 29
Motivation Applying Gr¨ obner-Bases Definition (Test Set) T is a test set for min { cy : Ay = b , y ∈ N n } if 1 all y ∈ T have negative cost ( cy < 0), 2 all y ∈ T solve Ay = 0 and 3 for each non-optimal solution y s ∈ N n of Ay s = b there is a y ∈ T s.t. y s + y ∈ N n . Optimization Algorithm 1 Find any solution y s ∈ N n of Ay s = b . 2 While ∃ y ∈ T : y s + y ∈ N n , let y s ← y s + y . 3 Return y s . Stephan Ritscher: Computation of Toric Gr¨ obner Bases 4/ 29
Motivation Applying Gr¨ obner-Bases Definition (Test Set) T is a test set for min { cy : Ay = b , y ∈ N n } if 1 all y ∈ T have negative cost ( cy < 0), 2 all y ∈ T solve Ay = 0 and 3 for each non-optimal solution y s ∈ N n of Ay s = b there is a y ∈ T s.t. y s + y ∈ N n . Optimization Algorithm 1 Find any solution y s ∈ N n of Ay s = b . 2 While ∃ y ∈ T : y s + y ∈ N n , let y s ← y s + y . 3 Return y s . How can we find a finite (small) test set??? Stephan Ritscher: Computation of Toric Gr¨ obner Bases 4/ 29
Prerequisites Polynomial Ideals Notation • K [ X ] = ring of polynomials over K in variables X = { x 1 , . . . , x n } • I ⊂ K [ X ] is an ideal iff for all a , b ∈ I , r ∈ K [ X ] 1 a + b ∈ I and 2 ar ∈ I . • � f 1 , . . . , f s � = ideal generated by f 1 , . . . , f s ∈ K [ X ] Stephan Ritscher: Computation of Toric Gr¨ obner Bases 5/ 29
Prerequisites Polynomial Ideals Notation • K [ X ] = ring of polynomials over K in variables X = { x 1 , . . . , x n } • I ⊂ K [ X ] is an ideal iff for all a , b ∈ I , r ∈ K [ X ] 1 a + b ∈ I and 2 ar ∈ I . • � f 1 , . . . , f s � = ideal generated by f 1 , . . . , f s ∈ K [ X ] Example • Q [ x 1 , x 2 , x 3 ] ∋ x 2 1 − x 2 x 3 , x 1 x 2 2 x 3 − 1 , x 5 2 x 3 3 − 1 x 2 1 − x 2 x 3 , x 1 x 2 2 x 3 − 1 , x 5 2 x 3 ∋ x 1 − x 3 2 x 2 • � � 3 − 1 3 = x 2 2 x 3 ( x 2 1 − x 2 x 3 ) − x 1 ( x 1 x 2 2 x 3 − 1) Stephan Ritscher: Computation of Toric Gr¨ obner Bases 5/ 29
Prerequisites Monomial Orderings Definition A total ordering ≺ of the monomials x α = x α 1 1 · · · x α n is admissible iff for n all α, β, γ ∈ N n (0 ∈ N ) 1 x α ≺ x β ⇒ x α x γ ≺ x β x γ and 2 1 ≺ x α for x α � = 1. LM ( f ) = largest monomial of f ∈ K [ X ] wrt. ≺ Stephan Ritscher: Computation of Toric Gr¨ obner Bases 6/ 29
Prerequisites Monomial Orderings (2) Example • Lexicographic Ordering: x α ≺ x β iff first nonzero entry of α − β is negative. x 1 ≻ x 3 2 x 2 3 ≻ x 2 2 x 3 3 • Graded Reverse Lexicographic Ordering: x α ≺ x β iff deg( x α ) < deg( x β ) or deg( x α ) = deg( x β ) and last nonzero entry of α − β is positive. x 3 2 x 2 3 ≻ x 2 2 x 3 3 ≻ x 1 Stephan Ritscher: Computation of Toric Gr¨ obner Bases 7/ 29
Prerequisites Monomial Orderings (3) Definition (Matrix Orderings) Given a matrix C ∈ K s , n , let x α ≺ x β iff the first nonzero entry of C α − C β is positive. Stephan Ritscher: Computation of Toric Gr¨ obner Bases 8/ 29
Prerequisites Monomial Orderings (3) Definition (Matrix Orderings) Given a matrix C ∈ K s , n , let x α ≺ x β iff the first nonzero entry of C α − C β is positive. Notes Any admissible monomial ordering can be • represented by a matrix C ∈ R n , n . • approximated up to an arbitrary, fixed degree by a matrix C ∈ Z n , n . Common monomial orderings can be represented by a matrix C ∈ Z n , n Stephan Ritscher: Computation of Toric Gr¨ obner Bases 8/ 29
Prerequisites Monomial Orderings (3) Definition (Matrix Orderings) Given a matrix C ∈ K s , n , let x α ≺ x β iff the first nonzero entry of C α − C β is positive. Notes Any admissible monomial ordering can be • represented by a matrix C ∈ R n , n . • approximated up to an arbitrary, fixed degree by a matrix C ∈ Z n , n . Common monomial orderings can be represented by a matrix C ∈ Z n , n Example ( n = 3) 1 0 0 Graded Reverse- 1 1 1 Lexicographic 0 1 0 lexicographic 0 0 − 1 Ordering: 0 0 1 Ordering: 0 − 1 0 Stephan Ritscher: Computation of Toric Gr¨ obner Bases 8/ 29
Prerequisites Gr¨ obner Bases Definition G is a Gr¨ obner basis of I wrt. a monomial ordering ≺ iff 1 I = � G � and 2 � LM ( I ) � = � LM ( G ) � . Stephan Ritscher: Computation of Toric Gr¨ obner Bases 9/ 29
Prerequisites Gr¨ obner Bases Definition G is a Gr¨ obner basis of I wrt. a monomial ordering ≺ iff 1 I = � G � and 2 � LM ( I ) � = � LM ( G ) � . Example x 2 1 − x 2 x 3 , x 1 x 2 2 x 3 − 1 , x 5 2 x 3 � � Consider I = 3 − 1 and the lexicographic monomial ordering. x 2 1 − x 2 x 3 , x 1 x 2 2 x 3 − 1 , x 5 2 x 3 � � • G 1 = 3 − 1 is no Gr¨ obner basis of I since x 1 ∈ � LM ( I ) � but x 1 / ∈ � LM ( G 1 ) � x 2 1 − x 2 x 3 , x 1 x 2 2 x 3 − 1 , x 5 2 x 3 3 − 1 , x 1 − x 3 2 x 2 � � • G 2 = is a Gr¨ obner basis 3 of I . Stephan Ritscher: Computation of Toric Gr¨ obner Bases 9/ 29
Problem Statement Problem Statement Definition (Toric Ideals) Given a matrix A ∈ Z k , n , the associated toric ideal is defined as � x α − x β : α, β ∈ N n : α − β ∈ ker( A ) � I ( A ) = Stephan Ritscher: Computation of Toric Gr¨ obner Bases 10/ 29
Problem Statement Problem Statement Definition (Toric Ideals) Given a matrix A ∈ Z k , n , the associated toric ideal is defined as � x α − x β : α, β ∈ N n : α − β ∈ ker( A ) � I ( A ) = Example � � A = 1 − 3 5 x 5 2 x 3 3 − 1 , x 1 − x 3 2 x 2 � � ⇒ I ( A ) = 3 Stephan Ritscher: Computation of Toric Gr¨ obner Bases 10/ 29
Problem Statement Problem Statement Definition (Toric Ideals) Given a matrix A ∈ Z k , n , the associated toric ideal is defined as � x α − x β : α, β ∈ N n : α − β ∈ ker( A ) � I ( A ) = Example � � A = 1 − 3 5 x 5 2 x 3 3 − 1 , x 1 − x 3 2 x 2 � � ⇒ I ( A ) = 3 Task Given • matrix A ∈ Z k , n and • monomial ordering ≺ (as matrix) compute a Gr¨ obner basis of I ( A ) wrt. ≺ . Stephan Ritscher: Computation of Toric Gr¨ obner Bases 10/ 29
Overview of Algorithms Common Parts ”Blue Print” of an Toric Ideal Algorithm Input : Matrix A, monomial ordering ≺ defined by matrix C Output : Gr¨ obner basis of I ( A ) Calculate lattice basis B of ker Z ( A ) Compute Markov basis M of ker Z ( A ) resp. ideal basis F of I ( A ) Compute Gr¨ obner basis of I ( A ) Stephan Ritscher: Computation of Toric Gr¨ obner Bases 11/ 29
Overview of Algorithms Common Parts Lattice Basis Definition A lattice L is a set of the form L = Z v 1 + Z v 2 + . . . + Z v s ⊂ Z n . The set { v 1 , . . . , v s } is a lattice basis of L . Stephan Ritscher: Computation of Toric Gr¨ obner Bases 12/ 29
Overview of Algorithms Common Parts Lattice Basis Definition A lattice L is a set of the form L = Z v 1 + Z v 2 + . . . + Z v s ⊂ Z n . The set { v 1 , . . . , v s } is a lattice basis of L . Example � � A = 1 − 3 5 � T + Z � T � � ⇒ ker Z ( A ) = Z 2 − 1 − 1 1 2 1 Stephan Ritscher: Computation of Toric Gr¨ obner Bases 12/ 29
Overview of Algorithms Common Parts Lattice Basis Definition A lattice L is a set of the form L = Z v 1 + Z v 2 + . . . + Z v s ⊂ Z n . The set { v 1 , . . . , v s } is a lattice basis of L . Example � � A = 1 − 3 5 � T + Z � T � � ⇒ ker Z ( A ) = Z 2 − 1 − 1 1 2 1 Computation • Since A ∈ Z k , n , L = ker Z ( A ) is a lattice. 1. Triangulate A with unimodular operations (Hermite decomposition). 2. Lattice basis B of L = ker Z ( A ) can be read off the triangular form. Stephan Ritscher: Computation of Toric Gr¨ obner Bases 12/ 29
Overview of Algorithms Common Parts Markov Basis Let α + be defined by α + i = max { α i , 0 } and α − = ( − α ) + . Definition Given a lattice L = ker Z ( A ), B is a Markov basis of L iff J ( B ) = { x α + − x α − : α ∈ B } is an ideal basis of I ( A ). Stephan Ritscher: Computation of Toric Gr¨ obner Bases 13/ 29
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