Achim Bachem

POLYNOMIAL ALGORITHMS FOR COMPUTING THE SMITH AND HERMITE NORMAL FORMS OF AN INTEGER MATRIX

Recently, Frumkin [9] pointed out that none of the well-known algorithms that transform an integer matrix into Smith [16] or Hermite [12] normal form is known to be polynomially bounded in its runn...

On the RAS-algorithm

Given a nonnegative real (m, n) matrixA and positive vectorsu, v, then the biproportional constrained matrix problem is to find a nonnegative (m, n) matrixB such thatB=diag (x) A diag (y) holds for some vectorsx ∈ ℝm andy ∈ ℝn and the row (column) sums ofB equalui (vj)i=1,...,m(j=1,..., n). A solution procedure (called the RAS-method) was proposed by Bacharach [1] to solve this problem. The main disadvantage of this algorithm is, that round-off errors slow down the convergence. Here we present a modified RAS-method which together with several other improvements overcomes this disadvantage.ZusammenfassungSeiA eine reelle (m, n) Matrix undu, v positive Vektoren. Das nichtnegative Matrixproblem besteht nun in der Aufgabe, eine nichtnegative (m, n) Matrix zu bestimmen, so daßB=diag(x) A diag (y) für Vektorenx ∈ ℝm undy ∈ ℝn gilt undui(vj)i=1, ...,m (j=1,...,n) die Zeilen- und Spaltensummen vonB darstellen. Eine Lösungsmethode (RAS-Verfahren) wurde von Bacharach [1] vorgeschlagen. Ein wesentlicher Nachteil dieses Algorithmus ist die Verlangsamung der Konvergenzgeschwindigkeit durch Rundungsfehler. Hier schlagen wir einen modifizierten RAS-Algorithmus vor, der zusammen mit anderen Verbesserungen diesen Nachteil überwindet.

Minimum norm problems over transportation polytopes

We consider the problem of updating input-output matrices, i.e., for given (m,n) matrices A ⩾ 0, W ⩾ 0 and vectors u ϵ Rm, vϵRn, find an (m,n) matrix X ⩾ 0 with prescribed row sums Σnj=1Xij = ui (i = 1,…,m) and prescribed column sums Σmi=1Xij = vj (j = 1,…,n) which fits the relations Xij = Aij + λiWij + Wij + Wijμj for all i,j and some λϵRm, μϵRn. Here we consider the question of existence of a solution to this problem, i.e., we shall characterize those matrices A, W and vectors u,v which lead to a solvable problem. Furthermore we outline some computational results using an algorithm of [2].

Minimal Inequalities and Subadditive Duality

In this note, we use a duality theorem for mixed integer programs (first explicitly stated by Johnson (1973) for the one-row group problem) to characterize minimal inequalities. This characterization extends earlier results, which assumed either a rational or a bounded constraint set (Blair (1978), Jeroslow (1979), Johnson (1976)), by relaxing these assumptions either entirely or almost so. It also extends results given first by Gomory and Johnson (1969) for the group problem.

A characterization of minimal valid inequalities for mixed integer programs

A simple condition on the underlying subadditive function is shown to characterize minimal valid inequalities. This result is proved in a very general master problem framework and completes the characterization there. We explain the condition also in the context of value functions and finally give some related, unresolved questions.

The Theorem of Minkowski for Polyhedral Monoids and Aggregated Linear Diophantine Systems

We study polyhedral monoids of the form M = {xeZn / Ax ≤ 0} for (m, n) integer matrices with rank m and prove in an elementary and constructive way that M has a finite basis, i.e. every x#x03B5;M is the nonnegative integer linear combination of a finite set of vectors. We show that this theorem holds also for monoids M(N, B)={xɛZ + S / Nx + By=o, yɛZ>n}. We consider the aggregated system GNx+GBy=o where G is an (r,m) aggregation matrix and show how the cardinality of a span of M(GN,GB) and M(N,B) relate to each other. Moreover we show how the group order of the Gomory group derived from M(N,B) changes if we aggregate Nx+By=o to GNx+GBy=o.

Primal and Dual Methods for Updating Input-Output Matrices

In this paper we consider the problem of updating input-output matrices, which is also known as the “constrained matrix problem” (cf. Bacharach [1]). Although Leontief developed his model for analyzing structural changes of an economy, his techniques can also be used to study various problems in management science. For instance, Gozinto graph models which are used in production theory, the Pichler model and various other models in production planning are based on modifications and extensions of the Leontief model (cf. Kloock [8], Lauenstein-Tempel [9], Seidel [12], Vogel [16]).

Applications of Polynomial Smith Normal form Calculations

An integer Square matrix with a deteminant of + 1 or −1 is called unimodular. Given a (m,m) integer matrix A, there exist unimodular matrices U,K such that S(A)=UAK is a diagonal matrix with positive diagonal elements d1,...,dr (r:=rank(A)) and zero diagonal elements dr+1,...,dm. In particular di divides di+1 (i=1,...,r−1). This was proved by Smith [21] in 1861 and the matrix S(A) is known as the Smith normal form of A.

About the XIth International Symposium on Mathematical Programming

The XIth International Symposium on Mathematical Programming was held under the auspices of the President of the Federal Republic of Germany Professor Dr. Karl Carstens.

Anhang: Vektor- und Matrizenrechnung

Wir haben in diesem Beitrag nur solche Ergebnisse der Vektor- und Matrizenrechnung verwendet, die eine elegante und ubersichtliche Darstellung unserer Satze erlauben.