40 research outputs found
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.
A polynomial-time algorithm for optimizing over N-fold 4-block decomposable integer programs
In this paper we generalize N-fold integer programs and two-stage integer
programs with N scenarios to N-fold 4-block decomposable integer programs. We
show that for fixed blocks but variable N, these integer programs are
polynomial-time solvable for any linear objective. Moreover, we present a
polynomial-time computable optimality certificate for the case of fixed blocks,
variable N and any convex separable objective function. We conclude with two
sample applications, stochastic integer programs with second-order dominance
constraints and stochastic integer multi-commodity flows, which (for fixed
blocks) can be solved in polynomial time in the number of scenarios and
commodities and in the binary encoding length of the input data. In the proof
of our main theorem we combine several non-trivial constructions from the
theory of Graver bases. We are confident that our approach paves the way for
further extensions
A finiteness theorem for Markov bases of hierarchical models
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 beyond a computable bound, the Markov
bases consist of elements of 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 tables.Comment: 13 pages, 1 figur