SI GILL
U
U
TA
IS
Marcus Kaiser
NAE
T
V E R S I TA
M
NI
Algebraic Methods in Integer and Mixed-Integer Programming
AUGU
S
Center for Mathematical Sciences, Technische Universität München
Chair of Applied Geometry and Discrete Mathematics
Abstract
This thesis deals with a primal approach to integer and mixed-integer linear programming. Test sets given by Graver bases are considered for both of these cases (cf. [1]).
Furthermore, Gröbner bases are applied to integer programs yielding another kind of test sets. The underlying algebraic concepts are introduced and algorithms for the computation
and subsequent application are given and examined.
The Situation
Test Sets based on Gröbner Bases
The considered problem is an integer linear program. Applications can be found in
production planning, scheduling, discrete tomography and combinatorial optimization.
Let m, n ∈ N be the dimensions, A ∈ Zm×n be the matrix and b ∈ Zm the vector
defining the linear constraints of the feasible region
The link between an integer linear program and the algebraic concept of Gröbner bases
is given by a special kind of ideals. For the matrix A the toric ideal IA is generated
by all the binomials corresponding to elements in the kernel of A.
+
−
IA := xu − xu : u ∈ ker(A) ∩ Zn
PA,b := {z ∈ Zn : Az = b, z ≥ 0} .
For a linear objective function given by a vector c ∈
of a solution to
min cT z
(ILPA,b,c )
Rn+ ,
the goal is the computation
subject to z ∈ PA,b .
Thereby, the condition of non-negativity on the cost vector c is not restricting if PA,b
is bounded.
The essential idea of the examined approach is the following. Starting with an initial
feasible solution an augmentation algorithm is applied. Iteratively a direction is
determined and a multiple of it is added to the current feasible solution in order to
reach another one. If for all non-optimal solutions such an augmentation step can be
performed and yields a better feasible solution with respect to the objective function the
optimal solution is finally reached for bounded programs. A set of vectors containing
such directions for every right hand side b and non-optimal feasible solution is called
a test set.
c
Gröbner Bases
Gröbner bases are special generating sets of ideals in the multivariate polynomial
ring K[x1 , . . . , xn ] over a field K. They represent a powerful tool in computer algebra and are based on the notion of monomial orderings. Monomial orderings are
total well-orderings on the set of monomials in K[x1 , . . . , xn ] which are in addition
compatible with the multiplication by a monomial.
With respect to such an ordering the term of a polynomial p ∈ K[x1 , . . . , xn ] which
contains the largest monomial can be considered. It is called the leading term of p
and denoted by LT (p).
A Gröbner basis G ⊆ I of an ideal I ⊆ K[x1 , . . . , xn ] is a generating set which allows
for all elements f ∈ I of the ideal the representation
f=
k
X
pi · gi ,
LT (pi · gi ) LT (f ) for all i ∈ [k]
Based on this definition, the toric ideal contains exactly the binomials with exponents
that differ by an element in the kernel of A. In particular, the binomials whose exponents are two feasible solutions to ILPA,b,c for any right hand side b lie in IA . It is not
trivial to compute a finite generating set of such an ideal given by the matrix A.
There exit various approaches. The fastest algorithm at the moment involves the idea
of project-and-lift (cf. [3]).
Also the objective function can be translated to the algebraic context. For this purpose
fix a matrix C ∈ Rr×n
which contains cT as the first row and additionally fulfils the
+
condition ker(C) ∩ Zn = {0}. For two vectors u, v ∈ Zn+ define
xu ≺C xv
∃ i ∈ [r] : (Cu)i < (Cv)i and (Cu)j = (Cv)j ∀ j < i .
:⇔
The relation C represents a monomial ordering refining the objective function in the
sense of the implication for all u, v ∈ Zn+
cT u < cT v
⇒
xu ≺C xv .
Applying Buchberger’s algorithm to a generating set of the toric ideal which consists
of binomials preserves this form. Hence, it yields a finite subset G ⊆ ker(A) ∩ Zn
+
−
such that the corresponding set of binomials {xu − xu : u ∈ G} is a Gröbner basis
with respect to C and u+ C u− for all u ∈ G. This computation can be done by
a geometric version of Buchberger’s Algorithm, which is an adaption to the presented
situation. It exploits the special structure of toric ideals in order to achieve a better
performance (cf. [3]).
An initial feasible solution to the problem can be gained by a proceeding similar to
the Phase-I of the Simplex algorithm. In addition, this produces a generating set of the
toric ideal. This approach, however, is not very efficient as Gröbner basis computations
are sensitive to the number of variables. There is a bound on the degree in a minimal
Gröbner basis which is doubly exponential in the dimension and cannot be improved
in general.
The set G constitutes a test set in a broader sense. Let z ∗ be the point in PA,b such
∗
that xz is minimal with respect to the constructed monomial order C . On the other
hand, take a feasible solution z such the corresponding monomial is not minimal. As
already mentioned the binomial given by this feasible points lies in the toric ideal and
therefore has a representation of the form
z
z∗
x −x =
k
X
u+
i
pi (x) · x
−x
u−
i
,
LTC pi (x) · x
u+
i
−x
u−
i
C LTC xz − xz
∗
i=1
where k ∈ N, pi ∈ K[x1 , . . . , xn ], ui ∈ G for all i ∈ [k].
This expression implies the
−
+
+
∗
existence of a j ∈ [k] with xuj = LTC xuj − xuj | LTC (xz − xz ) = xz . By
the inequality xz−uj ≺C xz , the feasible solution z − uj is better with respect to the
ordering given by C .
This yields the desired property of the set G. Any feasible non-optimal solution to
ILPA,b,c can be augmented by elements in G such that the resulting sequence is
strictly decreasing with respect to the monomial order C . Since this is a well-ordering
the minimal point z ∗ is finally reached. The construction of C assures that z ∗ is also
an optimal solution to the problem ILPA,b,c .
i=1
with k ∈ N, pi ∈ K[x1 , . . . , xn ], gi ∈ G for all i ∈ [k].
By Buchberger’s S-pair criterion, this representation is already possible for all elements in the ideal if the S-polynomial of two polynomials in the generating set can
be expressed in this special form. Thereby, a S-polynomials of two polynomials is a
linear combination of them wherein the leading terms of the summands cancel. From
this finite criterion a completion algorithm for the computation of Gröbner bases can
be constructed. Starting with a generating set of an ideal Buchberger’s algorithm
successively adds polynomials until this condition is satisfied (cf. [2]).
References
(1) R. Hemmecke. “On the positive sum property and the computation of Graver test sets”. In: Math. Program. 96.2, Ser. B (2003).
Algebraic and geometric methods in discrete optimization, pp. 247–269.
(2) D. Cox, J. Little, and D. O’Shea. Ideals, varieties, and algorithms. Third. Undergraduate Texts in Mathematics. An introduction to
computational algebraic geometry and commutative algebra. New York: Springer, 2007, pp. 49-114.
(3) J. A. De Loera, R. Hemmecke, and M. Köppe. Algebraic and geometric ideas in the theory of discrete optimization. Vol. 14. MOS-SIAM
Series on Optimization. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM), 2013, pp. 217–234.