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Groebner Bases, Toric Ideals and Integer Programming: An Application to Economics
Tan Tran
Junior
Major-Economics& Mathematics
History
• Groebner bases were developed by Buchberger in 1965, who later named them after his advisor, Wolfgang Groebner.
• Groebner bases were first used to solve the Ideal Membership Problem.
• Groebner bases can be described as special generating subsets of an ideal in a polynomial ring.
• They provide the foundation for many algorithms in algebraic geometry and commutative algebra.
Efficient Generation of Polynomial Ideals and an Application to Integer Programming
• Polynomial rings (in a finite number of indeterminates over a field) are the most fundamental objects of Computational Commutative Algebra.
• Polynomial rings are associative, commutative rings with identity.
1,....., nk x x
Ideals
• A subset of a ring, , is an ideal if it satisfies:
i. .
ii. If , then .
iii. If and , then .
Different examples of ideals in the polynomial ring: – Ideal generated by a finite number of polynomials.
– Ideal generated by monomials (LT ideal)
– Ideal generated by binomials (toric ideal).
– Ideal generated by varieties.
1,....., nI k x x
0 f,f g I f g I
f I 1,...., nh k x x hf I
Ideal Membership Problem
In the single variable case:
• Given , and an ideal I= , how to determine whether a given polynomial in
lies, for example, in ?
• How to find the generator of the ideal contained in k[x]?
• GCD is the key to the ideal membership problem. In the single variable case, if the division of the polynomial yields zero as the remainder, then it is in the given ideal.
4 61, 1x x
f k x 1,....., sf f
f
[ ]k x
k x
Polynomials of one variable • Given a nonzero polynomial , let
, where
• Division algorithm- let k be a field and let g be a nonzero polynomial in k[x]. Then every f in k[x] can be written as
f= qg + r, where q, r in K[x], and are unique.
0< deg(r) < deg (g)
Ex:
f k x
1
0 1 ....m m
mf a x a x a 0
mLT f a x
3 2 2 31 12 4 4
2 1 2 1 1x x x x x x x
Greatest Common Divisor and The Euclidean Algorithm
• The greatest common divisor of polynomials is a polynomial h such that:
1. h divides 2. if p is another polynomial which divides , then p divides
h.
• GCD gives the generator for the ideal . It exists
and is unique, as the consequence of the division algorithm.
• The Euclidean Algorithm is the classic algorithm for computing the GCD of two polynomials in k[x]. It can also be modified to calculate the GCD of multivariable polynomials.
1,...., sf f k x
1,...., sf f
1,...., sf f
1,...., sf f
Multivariable Polynomials
• A polynomial in K[ ] with coefficients in K is a finite linear combination of monomials.
–
–
–
–
1,...., nx x
,f a x a k
0deg max nmulti f
deg( )multi fLC f a k
deg( )( ) multi fLM f x
( ) ( ) ( )LT f LC f LM f
f
Monomial
• A monomial in K[ ] is a product of the form denoted and is the exponent vector with
• The total degree of this monomial is the sum
1,...., nx x1 2
1 2 ..... n
nx x x ax
1,...., na
0ia Z
1 ..... n
Term Ordering
• A monomial (term) order on is a relation on a set of monomials or, equivalently, on the set of exponent vectors a
, such that:
a) is a total order;
b) If
c) is a well ordering.
• Given a term order , every nonzero polynomial has a unique leading monomial denoted .
1,...., nk x xax
1,...., na 0ia Z
u v u w v wx x x x 0
nw
1,...., nf k x x
( )in f
Example
How to order ?
1. With respect to the lex order, the reordering of the terms of in decreasing order is:
2. With respect to grlex order:
3. With respect to grevlex order:
2 2 3 2 24 4 5 7 [ , , ]f xy z z x x z k x y z
3 2 2 2 25 7 4 4f x x z xy z z
2 2 2 3 27 4 5 4f x z xy z x z
2 2 2 3 24 7 5 4f xy z x z x z
f
Lexicographic Order
• Let and . We say if, in the vector difference
the left most nonzero entry is positive. We will write if .
Ex: since
1( ,...., )n 1 0( ,..... ) n
n
lex n
x x lex
3 2 4 3 2 1
1 2 3 1 2 3lexx x x x x x
(3,2,4) (3,2,1)lex (0,0,3)
Division algorithm in multivariable polynomials
• Fix a monomial order on and let be an ordered s-tuple of polynomials in . Then every f in can be written as
Where:
1. The
2. No term of r is divisible by any .
3. 1< i < s.
0
n
1 1 ... s sF a f a f r
1( ,...., )sF f f
1,..., nK x x
1, K ,...,i na r x x
( )iin f
( ) ( )i iin f in a f
1K ,..., nx x
• Changing the order of polynomials in the division set can change the result and, in particular, the remainders will be different.
EX:
Division 1:
Division 2:
2 2 1, 1xy x xy y
2
1 21, 1f xy f y
2 2( 1) 0 ( 1) ( )xy x y xy y x y
r x y
2
1 21, 1f y f xy
2 2( 1) 0 ( 1) 0xy x x y xy
0r
• Ideal Membership problem: Given and an ideal , determine if
• It is impossible to use the method of the one variable case to solve the ideal membership problem.
1,...., sI f f f I1K[ , , ]nf x x
Groebner Bases
Definition: Given a monomial order and an ideal . We say that is a Groebner Basis of I if .
• Any Groebner basis for an ideal I is a generating set for I.
1[ ,...., ]nI F x x 1,...., tg g
1( ),..., ( ) ( )tin g in g in I
Groebner Bases
• If is a Groebner basis for I, and
, then f can be written uniquely in the form f= g+r, where and no term of r is divisible by .
• If is a Groebner basis for I, and , then the remainder of f on division by is zero.
1,...., tg g
1K[ ,..., ]nf x x
1 1( ... )t tg I g a g a g
( )iin g
1{ ,...., }tg g
1K[ ,..., ]nf x x f I
1,...., ng g
Hilbert Basis Theorem
• Every ideal has a finite generating set. That is, for some
• Fix a monomial order. Then every ideal other than {0} has a finite Groebner basis.
1,...., nI k x x
1,...., tI g g 1,...., tg g I
1[ ,...., ]nI K x x
Buchberger Algorithm
Let be nonzero. Fix a monomial order and let , . Let
be the least common multiple of and .
The S polynomial of f and g is:
( , )( ) ( )
x xS f g f g
LT f LT g
1, K[ ,..., ]nf g x x
( ) , ( )u vin f cx in g dx ,c d k
wx ux vx
Example
Which is not divisible by or by .
3 2f x xy
2 22g x y y x 3 3
3 2 2
3 2( , ) ( 2 ) ( 2 )
x y x yS f g x xy x y y x
x x y
4 2 3 3 43 3
3 2 2
2 2x y x y x yx y x y
x x y x y
2 2 2 22 2xy xy x x
( )in f ( )in g
Buchberger Algorithm
Input: Output: a Groebner basis for I, with
REPEAT FOR each pair in G’ DO
IF THEN UNTIL
is the unique remainder of f on division by
(r is normal form)
1( ,...., )sF f f
1( ,...., )sG g g F G' :G F
' :G G{ , },p q p q
0S : { }G G S
'
: ( , )G
S S p q
'G GG
r f1{ ,...., }nG g g
Example
REPEAT
REPEAT
3 2 2 2
1 { 2 , 2 , }G x xy x y y x x
1
1 2( , ) 0G
S g g
1
1 3 4( , ) 2G
S g g xy g
1 2
2 3 5( , ) 2G
S g g x y g 3 2 2 2 2
2 { 2 , 2 , , 2 , 2 }G x xy x y x x xy x y
2 3
1 5( , ) 4G
S g g y
2 3
4 5( , ) 2G
S g g y
2
( , ) 0G
i jS g g
3 2 2 2 2 3
3 { 2 , 2 , , 2 , 2 , }G x xy x y x x xy x y y
3
( , ) 0G
i jS g g
all other i,j
3G G
i j
My Research
• My research consists of applying specific concepts and constructions from Commutative Algebra to the area known in Economics as Integer Programming.
• In particular, by means of several algorithms (Conti-Traverso, Buchberger), a specific Transportation Problem (one of the classical problems of Integer Programming) is solved.
• To find the Toric Ideal and Groebner Basis needed to obtain the optimal solution, it was necessary to explore several versions of Computational Commutative Algebra software such as CoCoa and Maucalay.
Terminologies
• A homomorphism from a ring (R, +, ·) to a ring (S, ‡, *) is a function f from R to S that that preserves the structure of a ring, that is, for all a, b in R, the following identities hold:
• f(a + b) = f(a) ‡ f(b)
• f(a · b) = f(a) * f(b)
Toric Ideal
Fix
Identify each with a monomial (Laurent polynomial ring)
Consider the semigroup homomorphism:
Which lifts to:
ker( ) is , the Toric Ideal of A
1{ ,...., } d
nA a a Z
ia1 1
1 1[ ,...., , ,..., ]ia
d dt F t t t t
: n dN Z
1 1 1( ,..., ) ....n n nu u u u u aa
^1 1
1 1 1: [ ,...., ] [ ,...., , ,...., ]n d dk x x k t t t t
AI^
Toric Ideal
• Toric ideal is a special class of ideals in the polynomial ring.
• The toric ideal is a spanned as a F vector space by the set of binomials
{ : , ( ) ( )}u v nx x u v N and u v
AI
Integer Programming
• Given any “cost vector” , find a point u in which minimizes the value of the linear functional .
• The idea here is to apply toric ideals to model and solve the problem.
nw R1( )b
u u w
Given any cost vector find a point u in which minimizes the value of the linear functional.
The fiber consists of all non-negative integer matrices with row sums r and column sums c.
nw R1( )b
1
' '
1
'
1 1
1 1 1 1
Fix , ,
,....,
,..., , ,
{ : 1,....., , 1,...., }
: ,
{ } ( ,...., ; ,...., )
s
s
t
t
d
i j
s t s t
t t s s
ij j sj i it
j j i i
s t Z
e e N
e e N n s t d s t
A e e i s j t N
N N
u u u u u
u u w
1( ; )r c s t
Algorithm
Input: a matrix A and a cost function
Output: An optimal point with minimal for any given .
1. Compute the reduced Groebner basis G for with respect to weight order.
2. For any given vector do: a. Find any feasible solution .
b. Compute the normal form with respect to the weight order. Output u.
d nnw R
1( )u b u w( )b im
( )b im 1( )v b
of u vx x
The Transportation Problem
s= 4 factories
t=3 stores
produces 120 units demands 183 units
produces 204 units demands 190 units
produces 92 units demands 98 units
produces 55 units
W is the non negative real cost associated with transporting one unit from to .
We want the transportation plan that minimizes total cost of shipping all 942 units. In the fiber:
We have all the possible transportation plans.
1 2 3 4, , ,F F F F
1 2 3, ,S S S
1F
2F
4F3F
1S
2S
3S
jSiF
1(120,204,92,55;183,190,98)
And let be the diagonal term order .
In this case is the set of 2*2 minors.
Consider the feasible solution
Corresponding to the matrix v =
11 12( .... )stx x x
120 0 0 14 190 0 0 0 92 49 0 6
11 12 13 21 22 23 31 32 33 41 42 43
vx x x x x x x x x x x x x
wG
1 1 3
2 1 1
1 1 2
3 2 1
w
120 0 0
14 190 0
0 0 92
49 0 6
The normal form of with respect to the Groebner basis
is
Which is the optimal solution
vx
{ :1 4,1 4}il jk ik jlx x x x i j k l
120 161 43 63 29 55
11 22 23 31 32 43
ux x x x x x x
120 0 0
0 161 43
63 29 0
0 0 55
u
Use of Software
• CoCoa
• MaCaulay
CONCLUSIONS
• The use of techniques from Groebner Basis Theory, and Commutative Algebra in general, to solve a classical problem from Economics, Integer Programming, shows how advances in research in pure mathematics find applications never imagined. As I am a double major, this project let me see first hand how a very abstract area of Mathematics has an application to my other field.
