Abstract. Any configuration of lattice vectors gives rise to a hierarchy of higher-dimensional configurations which generalize the Lawrence construction in geometric combinatorics. We prove finiteness results for the Markov bases, Graver bases and facet posets of these configurations, and we discuss applications to the statistical theory of log-linear models. 1.

We show that the complexity of the Markov bases of multidimensional tables stabilizes eventually if a single table dimension is allowed to vary. In particular, if this table dimension is greater than a computable bound, the Markov bases consist of elements from Markov bases of smaller tables. We give an explicit formula for this bound in terms of Graver bases. We also compute these Markov and Graver complexities for all K × 2 × 2 × 2 tables. 1

this paper we give some basic characterizations of minimal Markov basis for a connected Markov chain, which is used for performing exact tests in discrete exponential families given a sufficient statistic. We also give a necessary and sufficient condition for uniqueness of minimal Markov basis. A general algebraic algorithm for constructing a connected Markov chain was given by Diaconis and Sturmfels (1998). Their algorithm is based on computing Grobner basis for a certain ideal in a polynomial ring, which can be carried out by using available computer algebra packages. However structure and interpretation of Grobner basis produced by the packages are not necessarily clear, due to the lack of symmetry and minimality inherent in Grobner basis computation. Our approach clarifies partially ordered structure of minimal Markov basis

We consider testing the quasi-independence hypothesis for two-way contingency tables which contain some structural zero cells. For sparse contingency tables where the large sample...

Abstract. Data swapping, a term introduced in 1978 by Dalenius and Reiss for a new method of statistical disclosure protection in confidential data bases, has taken on new meanings and been linked to new statistical methodologies over the intervening twenty-five years. This paper revisits the original (1982) published version of the the Dalenius-Reiss data swapping paper and then traces the developments of statistical disclosure limitation methods that can be thought of as rooted in the original concept. The emphasis here, as in the original contribution, is on both disclosure protection and the release of statistically usable data bases.

In this paper we define an invariant Markov basis for a connected Markov chain over the set of contingency tables with fixed marginals and derive some characterizations of minimality of the invariant basis. We also give a necessary and sufficient condition for uniqueness of invariant minimal Markov basis. The invariance here refers to permutation of indices of each axis of the contingency tables. If the categories of each axis do not have any order relations among them, it is natural to consider the action of the symmetric group on each axis of the contingency table. A general algebraic algorithm for obtaining a Markov basis was given by Diaconis and Sturmfels (1998). Their algorithm is based on computing Gröbner basis of a well-specified polynomial ideal. However the reduced Gröbner basis depends on the particular term order and is not symmetric. Therefore it is of interest to consider properties of invariant Markov basis. We study minimality of invariant Markov basis using techniques of Takemura and Aoki (2003).

Abstract not found

this paper we present indispensable moves of Markov bases for connected Markov chains over three-way contingency tables with fixed two-dimensional marginals. In Aoki and Takemura (2003a) we proved that there exists a unique minimal basis for 3 contingency tables consisting of four types of indispensable moves. Generalizing this result, we present a list of indispensable moves of the unique minimal Markov basis for 3 4 contingency tables. This list allows us to actually perform exact tests of no three-factor interaction in three-way tables of these sizes. There are 21 types of indispensable moves for the 3 K case and 14 types of indispensable moves for the 444 case. A proof of the fact that these indispensable moves form the unique minimal basis along the lines of Aoki and Takemura (2003a) is unfortunately too long and omitted. In addition we give a (non-exhaustive) list of indispensable moves for larger three-way tables. In this paper we prove some results on constructing indispensable moves from other indispensable moves. Our indispensable moves for larger tables were found by using these results combined with some computer searches. Closely connected notions to indispensability are the notions of fundamental moves and circuits discussed in Ohsugi and Hibi (1999, 2003). We also indicate whether our indispensable moves are fundamental or circuits. 1

Abstract. Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject to integrality requirements for the variables. This chapter is dedicated to this topic. The primary goal is a study of a simple version of general nonlinear integer problems, where all constraints are still linear. Our focus is on the computational complexity of the problem, which varies significantly with the type of nonlinear objective function in combination with the underlying combinatorial structure. Numerous boundary cases of complexity emerge, which sometimes surprisingly lead even to polynomial time algorithms. We also cover recent successful approaches for more general classes of problems. Though no positive theoretical efficiency results are available, nor are they likely to ever be available, these seem to be the currently most successful and interesting approaches for solving practical problems. It is our belief that the study of algorithms motivated by theoretical considerations

We give a description of the minimal primes of the ideal generated by the 22 adjacent minors of a generic matrix. We also compute the complete prime decomposition of the ideal of adjacent mm minors of an mn generic matrix when the characteristic of the ground eld is zero. A key intermediate result is the proof that the ideals which appear as minimal primes are, in fact, prime ideals. This introduces a large new class of mixed determinantal ideals that are prime.