565,138 research outputs found
The Complexity of Three-Way Statistical Tables
Multi-way tables with specified marginals arise in a variety of applications
in statistics and operations research. We provide a comprehensive complexity
classification of three fundamental computational problems on tables:
existence, counting and entry-security.
One major outcome of our work is that each of the following problems is
intractable already for "slim" 3-tables, with constant and smallest possible
number 3 of rows: (1) deciding existence of 3-tables with given consistent
2-marginals; (2) counting all 3-tables with given 2-marginals; (3) finding
whether an integer value is attained in entry (i,j,k) by at least one of the
3-tables satisfying given (feasible) 2-marginals. This implies that a
characterization of feasible marginals for such slim tables, sought by much
recent research, is unlikely to exist.
Another important consequence of our study is a systematic efficient way of
embedding the set of 3-tables satisfying any given 1-marginals and entry upper
bounds in a set of slim 3-tables satisfying suitable 2-marginals with no entry
bounds. This provides a valuable tool for studying multi-index transportation
problems and multi-index transportation polytopes
Higher Lawrence configurations
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 face posets of these configurations, and we discuss
applications to the statistical theory of log-linear models.Comment: 12 pages. Changes from v1 and v2: minor edits. This version is to
appear in the Journal of Combinatorial Theory, Ser.
The Geometry of Statistical Models for Two-Way Contingency Tables with Fixed Odds Ratios
We study the geometric structure of the statistical models for two-by-two
contingency tables. One or two odds ratios are fixed and the corresponding
models are shown to be a portion of a ruled quadratic surface or a segment.
Some pointers to the general case of two-way contingency tables are also given
and an application to case-control studies is presented.Comment: References were not displaying properly in the previous versio
Combinatorics and Geometry of Transportation Polytopes: An Update
A transportation polytope consists of all multidimensional arrays or tables
of non-negative real numbers that satisfy certain sum conditions on subsets of
the entries. They arise naturally in optimization and statistics, and also have
interest for discrete mathematics because permutation matrices, latin squares,
and magic squares appear naturally as lattice points of these polytopes.
In this paper we survey advances on the understanding of the combinatorics
and geometry of these polyhedra and include some recent unpublished results on
the diameter of graphs of these polytopes. In particular, this is a thirty-year
update on the status of a list of open questions last visited in the 1984 book
by Yemelichev, Kovalev and Kravtsov and the 1986 survey paper of Vlach.Comment: 35 pages, 13 figure
- …