178 research outputs found
Combinatorial degree bound for toric ideals of hypergraphs
Associated to any hypergraph is a toric ideal encoding the algebraic
relations among its edges. We study these ideals and the combinatorics of their
minimal generators, and derive general degree bounds for both uniform and
non-uniform hypergraphs in terms of balanced hypergraph bicolorings,
separators, and splitting sets. In turn, this provides complexity bounds for
algebraic statistical models associated to hypergraphs. As two main
applications, we recover a well-known complexity result for Markov bases of
arbitrary 3-way tables, and we show that the defining ideal of the tangential
variety is generated by quadratics and cubics in cumulant coordinates.Comment: Revised, improved, reorganized. We recommend viewing figures in colo
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
Noetherianity up to symmetry
These lecture notes for the 2013 CIME/CIRM summer school Combinatorial
Algebraic Geometry deal with manifestly infinite-dimensional algebraic
varieties with large symmetry groups. So large, in fact, that subvarieties
stable under those symmetry groups are defined by finitely many orbits of
equations---whence the title Noetherianity up to symmetry. It is not the
purpose of these notes to give a systematic, exhaustive treatment of such
varieties, but rather to discuss a few "personal favourites": exciting examples
drawn from applications in algebraic statistics and multilinear algebra. My
hope is that these notes will attract other mathematicians to this vibrant area
at the crossroads of combinatorics, commutative algebra, algebraic geometry,
statistics, and other applications.Comment: To appear in Springer's LNM C.I.M.E. series; several typos fixe
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