58 research outputs found
Finiteness theorems in stochastic integer programming
We study Graver test sets for families of linear multi-stage stochastic
integer programs with varying number of scenarios. We show that these test sets
can be decomposed into finitely many ``building blocks'', independent of the
number of scenarios, and we give an effective procedure to compute these
building blocks. The paper includes an introduction to Nash-Williams' theory of
better-quasi-orderings, which is used to show termination of our algorithm. We
also apply this theory to finiteness results for Hilbert functions.Comment: 36 p
N-fold integer programming in cubic time
N-fold integer programming is a fundamental problem with a variety of natural
applications in operations research and statistics. Moreover, it is universal
and provides a new, variable-dimension, parametrization of all of integer
programming. The fastest algorithm for -fold integer programming predating
the present article runs in time with the binary length of
the numerical part of the input and the so-called Graver complexity of
the bimatrix defining the system. In this article we provide a drastic
improvement and establish an algorithm which runs in time having
cubic dependency on regardless of the bimatrix . Our algorithm can be
extended to separable convex piecewise affine objectives as well, and also to
systems defined by bimatrices with variable entries. Moreover, it can be used
to define a hierarchy of approximations for any integer programming problem
Equality of Graver bases and universal Gr\"obner bases of colored partition identities
Associated to any vector configuration A is a toric ideal encoded by vectors
in the kernel of A. Each toric ideal has two special generating sets: the
universal Gr\"obner basis and the Graver basis. While the former is generally a
proper subset of the latter, there are cases for which the two sets coincide.
The most prominent examples among them are toric ideals of unimodular matrices.
Equality of universal Gr\"obner basis and Graver basis is a combinatorial
property of the toric ideal (or, of the defining matrix), providing interesting
information about ideals of higher Lawrence liftings of a matrix. Nonetheless,
a general classification of all matrices for which both sets agree is far from
known. We contribute to this task by identifying all cases with equality within
two families of matrices; namely, those defining rational normal scrolls and
those encoding homogeneous primitive colored partition identities.Comment: minor revision; references added; introduction expanded
Computing holes in semi-groups and its applications to transportation problems
An integer feasibility problem is a fundamental problem in many areas, such
as operations research, number theory, and statistics. To study a family of
systems with no nonnegative integer solution, we focus on a commutative
semigroup generated by a finite set of vectors in and its saturation. In
this paper we present an algorithm to compute an explicit description for the
set of holes which is the difference of a semi-group generated by the
vectors and its saturation. We apply our procedure to compute an infinite
family of holes for the semi-group of the transportation
problem. Furthermore, we give an upper bound for the entries of the holes when
the set of holes is finite. Finally, we present an algorithm to find all
-minimal saturation points of .Comment: Presentation has been improved according to comments by referees.
This manuscript has been accepted to "Contributions to Discrete Mathematics
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