Higher Lawrence configurations

Santos, Francisco; Sturmfels, Bernd

doi:10.1016/s0097-3165(03)00092-x

Cited by 71 publications

(109 citation statements)

References 9 publications

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“…On the other hand, if only one of the table dimensions is allowed to vary, then there is a bounded finite structure to the Markov bases. This theorem was first proven in [62] and generalizes a result in [81]. Theorem 1.2.18 (Long tables).…”

Section: Consider the Hypothesis Testing Problemmentioning

confidence: 53%

Lectures on Algebraic Statistics

Drton

Sturmfels

Sullivant

2009

Self Cite

332

423

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Section: Consider the Hypothesis Testing Problemmentioning

confidence: 53%

Lectures on Algebraic Statistics

Drton

Sturmfels

Sullivant

2009

Self Cite

332

423

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“…Two nice stabilization results established by Hoşten and Sullivant [71] and Santos and Sturmfels [113] immediately imply that if A 1 and A 2 are kept fixed, then the size of the Graver basis increases only polynomially in the number n of copies of A 1 and A 2 .…”

Section: Boundary Cases Of Complexitymentioning

confidence: 89%

“…Nonetheless, the technical difficulties when adjusting the proofs for the pure integer case to the mixed-integer situation can be overcome [68]. It should be noted, however, that the Graver basis of n-fold matrices does not show a stability result similar to the pure integer case as presented in [71,113]. Thus, we do not get a nice polynomial time algorithm for solving mixed-integer convex n-fold problems.…”

Section: Boundary Cases Of Complexitymentioning

confidence: 92%

Nonlinear Integer Programming

Li¹,

Sun²

2006

International Series in Operations Research &Amp; Management Science

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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 and those motivated by our desire to solve practical instances should and do inform one another. So it is with this viewpoint that we present the subject, and it is in this direction that we hope to spark further research.

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“…We purposely formulate them in rather general terms, as all of the problems pose challenges that are both of theoretical and computational nature, and we believe are relevant to the mathematical and statistical audience jointly. Eriksson et al [20] posed a related conjecture, and wondered whether some finite complexity properties of the facial structure of the marginal cone is related to the finite complexity properties of Markov bases proved in Santos and Sturmfels [36].…”

Section: Examplementioning

confidence: 99%

Algebraic Statistics and Contingency Table Problems: Log-Linear Models, Likelihood Estimation, and Disclosure Limitation

Dobra

Fienberg

Rinaldo

et al. 2008

Emerging Applications of Algebraic Geometry

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Abstract. Contingency tables have provided a fertile ground for the growth of algebraic statistics. In this paper we briefly outline some features of this work and point to open research problems. We focus on the problem of maximum likelihood estimation for log-linear models and a related problem of disclosure limitation to protect the confidentiality of individual responses. Risk of disclosure has often been measured either formally or informally in terms of information contained in marginal tables linked to a log-linear model analysis and has focused on disclosure potential of small cell counts, especially those equal to 1 or 2. One way to assess risk is to compute bounds for cell entries given a set of released marginals. Both of these methodologies become complicated for large sparse tables. This paper revisits the problem of computing bounds for cell entries and picks up on a theme first suggested in Fienberg [21] that there is an intimate link between the ideas on bounds and the existence of maximum likelihood estimates, and shows how these ideas can be made rigorous through the underlying mathematics of the same geometric/algebraic framework. We illustrate the linkages through a series of examples. We also discuss the more complex problem of releasing marginal and conditional information. We illustrate the statistical features of the methodology on two examples and then conclude with a series of open problems.

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Higher Lawrence configurations

Cited by 71 publications

References 9 publications

Lectures on Algebraic Statistics

Lectures on Algebraic Statistics

Nonlinear Integer Programming

Algebraic Statistics and Contingency Table Problems: Log-Linear Models, Likelihood Estimation, and Disclosure Limitation

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