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 k x,..., (in a finite 1 xn number of indeterminates over a field) are the most fundamental objects of Computational Commutative Algebra. Polynomial rings are associative, commutative rings with identity.
A subset of a ring, satisfies: Ideals i. 0 f. ii. If f, g I, then f g. I iii. If f I and h k x x, then., is an ideal if it 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. I k x,..., 1 xn,..., 1 n hf I
Ideal Membership Problem In the single variable case: Given f k x, and an ideal I= f,..., 1 fs, how to determine whether a given polynomial f in 4 6 kx lies, for example, in x 1, x 1? ] kx [ 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.
Polynomials of one variable Given a nonzero polynomial, let f a0x a1 x... a m m m1, where LT f a0x 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 k x m Ex: f= qg + r, where q, r in K[x], and are unique. 0< deg(r) < deg (g) 1 1 x 2x x 1 x x 2x 1 x 1 3 2 2 3 2 4 4
Greatest Common Divisor and The Euclidean Algorithm The greatest common divisor of polynomials f,..., is a polynomial h such that: 1 fs k x 1. h divides f,..., 1 fs 2. if p is another polynomial which divides f,..., 1 fs, then p divides h. GCD gives the generator for the ideal,..., 1 s. 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. f f
Multivariable Polynomials A polynomial f in K[ x,..., ] with coefficients 1 xn in K is a finite linear combination of monomials. f a x, a k n max 0 multi deg f multi deg( f ) LC f a k LM ( f ) x multi deg( f ) LT( f ) LC( f ) LM ( f )
Monomial A monomial in K[ x,..., 1 xn ] is a product of 1 2 the form... n x 1 x 2 x a n denoted x and a is the exponent vector with ai 1,..., n Z 0 The total degree of this monomial is the sum 1... n
Term Ordering A monomial (term) order on k x,..., 1 xn is a a relation on a set of monomials x or, equivalently, on the set of exponent vectors a a,..., 1 n, a such that: i Z 0 a) is a total order; b) If x x x x u v u w vw n w 0 c) is a well ordering. Given a term order, every nonzero polynomial f k x,..., 1 xn has a unique leading monomial denoted in ( f ).
Example 2 2 3 2 2 How to order? f 4xy z 4z 5x 7 x z k[ x, y, z] 1. With respect to the lex order, the reordering of the terms of f in decreasing order is: 3 2 2 2 2 f 5x 7x z 4xy z 4z 2. With respect to grlex order: f 7x z 4xy z 5x 4z 2 2 2 3 2 3. With respect to grevlex order: f 4xy z 7x z 5x 4z 2 2 2 3 2
Lexicographic Order n Let ( 1,..., n ) and ( 1,... n) 0. We say if, in the vector difference lex the left most nonzero entry is positive. We will write x x if lex. n Ex: (3,2,4) lex (3,2,1) since (0,0,3) 3 2 4 x 3 x 2 x 1 1 2 3 lex 1 2 3 x x x
Division algorithm in multivariable polynomials n Fix a monomial order on 0 and let F ( f1,..., f s ) be an ordered s-tuple of polynomials in K x,..., 1 xn. Then every f in K x x can be written as 1,..., n F a f... a f r 1 1 s s Where: 1. The ai, r K x1,..., xn 2. No term of r is divisible by any ( ). 3. 1< i < s. in ( f ) in( a f ) i i in f i
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 xy x xy 1, y 1 f xy f y 2 1 1, 2 1 2 2 xy x y xy y x y 2 1 2 ( 1) 0 ( 1) ( ) r x y f y 1, f xy 1 2 2 xy x x y xy ( 1) 0 ( 1) 0 r 0
Ideal Membership problem: Given 1 and an ideal I f f, determine if f I 1,..., s f K[ x,, x n ] It is impossible to use the method of the one variable case to solve the ideal membership problem.
Groebner Bases Definition: Given a monomial order and an ideal I F[ x1,..., x n ]. We say that g,..., 1 gt is a Groebner Basis of I if in ( g ),..., in ( g ) in ( I). 1 Any Groebner basis for an ideal I is a generating set for I. t
If g,..., 1 gt Groebner Bases is a Groebner basis for I, and K[ 1,..., ], then f can be written uniquely in the form f= g+r, where g I( g a1g 1... atgt) and no term of r is divisible by in ( g i ). f x x n { g,..., g t } If 1 is a Groebner basis for I, and f K[ x1,..., x n ], then f I the remainder of f on division by g,..., 1 gn is zero.
Hilbert Basis Theorem Every ideal set. That is, I k x,..., 1 xn I g,..., 1 gt has a finite generating for some g g I Fix a monomial order. Then every ideal I K[ x1,..., x n ] other than {0} has a finite Groebner basis. 1,..., t
Buchberger Algorithm Let f, g K[ x1,..., x n ] be nonzero. Fix a monomial order and let ( ) u v in f cx, in ( g) dx, c, d k. Let w x u v be the least common multiple of x and x. The S polynomial of f and g is: x x S( f, g) f g LT( f ) LT( g)
Example 3 f x 2xy 2 2 g x y 2y x 3 3 x y 3 x y 2 2 S( f, g) ( x 2 xy) ( x y 2 y x) 3 2 x x y 4 2 3 3 4 3 2x y 3 2x y x y x y x y 3 2 2 x x y x y 2xy 2xy x x 2 2 2 2 Which is not divisible by in ( f ) or by in ( g).
Buchberger Algorithm Input: Output: a Groebner basis G ' : F REPEAT F f1 f s G': G (,..., ) FOR each pair G g1 g s { p, q}, p q (,..., ) F G in G DO for I, with UNTIL r S : S( p, q) G' IF S 0 THEN G : G { S} G G' G f is the unique remainder of f on division by G { g (r is normal form) 1,..., g n }
G x xy x y y x x 3 2 2 2 1 { 2, 2, } S g g 1 (, ) 0 1 2 G G 1 S( g, g ) 2xy g 1 3 4 G 1 2 S( g, g ) x 2y g Example 2 3 5 3 2 2 2 2 G2 { x 2 xy, x 2 y x, x, 2 xy, x 2 y } S( g, g ) 4y 1 5 4 5 G G 2 3 S( g, g ) 2y S g g G 2 3 2 (, ) 0 i j REPEAT G x xy x y x x xy x y y 3 2 2 2 2 3 3 { 2, 2,, 2, 2, } S g g G 3 (, ) 0 G3 i j G all other i,j i j REPEAT
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 d A { a1,..., an} Z Identify each a i with a monomial a i 1 1 t F[ t1,..., td, t1,..., td ] (Laurent polynomial ring) Consider the semigroup homomorphism: : N Which lifts to: n Z d u ( u,..., u ) u a... u a 1 n 1 1 n n ^ ker( ) is : k[ x,..., x ] k[ t,..., t, t,..., t ] ^ 1 1 1 n 1 d 1 d I A, the Toric Ideal of A
Toric Ideal Toric ideal is a special class of ideals in the polynomial ring. The toric ideal I A is a spanned as a F vector space by the set of binomials { x u x v : u, v N n and( u) ( v)}
Integer Programming Given any cost vector w R n, find a point u in 1 () b which minimizes the value of the linear functional u u w. The idea here is to apply toric ideals to model and solve the problem.
1 () b n Given any cost vector w R find a point u in which minimizes the value of the linear functional. 1 Fix s, t Z, e,..., e N s ' ' t 1,..., t,, s e e N n s t d s t u ' { i j : 1,...,, 1,..., } st st : N N, u w t t s s { u } ( u,..., u ; u,..., u ) ij 1j sj i1 it j1 j1 i1 i1 The fiber 1 ( rc ; ) consists of all non-negative integer matrices with row sums r and column sums c. A e e i s j t N d st
Algorithm Input: a dnmatrix A and a cost function w R Output: An optimal point u 1 () b with uw minimal for any given bim( ). 1. Compute the reduced Groebner basis G for with respect to weight order. 2. For any given vector bim( ) do: a. Find any feasible solution v 1 () b. u v b. Compute the normal form x of x with respect to the weight order. Output u. n
The Transportation Problem s= 4 factories t=3 stores S, S, S F 1 F 2 F F 3 4 F, F, F, F 1 2 3 4 1 2 3 produces 120 units produces 204 units produces 92 units produces 55 units demands 183 units demands 190 units demands 98 units W is the non negative real cost associated with transporting one unit from to. F i S j We want the transportation plan that minimizes total cost of shipping all 942 units. In the fiber: 1 (120,204,92,55;183,190,98) S 1 S 2 S 3 We have all the possible transportation plans.
And let be the diagonal term order ( x11 x12... x st ). In this case is the set of 2*2 minors. G w 1 1 3 2 1 1 w 1 1 2 3 2 1 Consider the feasible solution v 120 0 0 14 190 0 0 0 92 49 0 6 x x11 x12 x13 x 21x22 x23x31 x32 x33 x41 x42x43 Corresponding to the matrix v = 120 0 0 14 190 0 0 0 92 49 0 6
The normal form of Groebner basis v x with respect to the { x x x x :1 i j 4,1 k l 4} il jk ik jl u x x x x x x x Which is the optimal solution 120 161 43 63 29 55 11 22 23 31 32 43 is u 120 0 0 0 161 43 63 29 0 0 0 55
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.