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- Serkan Hosten, Seth Sullivant
- J. Comb. Theory, Ser. A
- 2007

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… (More)

- Serkan Hosten, Bernd Sturmfels
- Combinatorica
- 2007

We determine the maximal gap between the optimal values of an integer program and its linear programming relaxation, where the matrix and cost function are fixed but the right hand side is unspecified. Our formula involves irreducible decomposition of monomial ideals. The gap can be computed in polynomial time when the dimension is fixed.

- Serkan Hosten, Bernd Sturmfels
- IPCO
- 1995

- Serkan Hosten, Seth Sullivant
- J. Comb. Theory, Ser. A
- 2002

This article studies the polyhedral structure and combinatorics of polytopes that arise from hierarchical models in statistics, and shows how to construct Gröbner bases of toric ideals associated to a subset of such models. We study the polytopes for cyclic models, and we give a complete polyhedral description of these polytopes in the binary cyclic case.… (More)

- Serkan Hosten, Amit Khetan, Bernd Sturmfels
- Foundations of Computational Mathematics
- 2005

Given a model in algebraic statistics and data, the likelihood function is a rational function on a projective variety. Algebraic algorithms are presented for computing all critical points of this function, with the aim of identifying the local maxima in the probability simplex. Applications include models specified by rank conditions on matrices and the… (More)

- Serkan Hosten, Jay Shapiro
- J. Symb. Comput.
- 2000

- Serkan Hosten, Walter D. Morris
- Discrete Mathematics
- 1999

A configuration of lattice vectors is supernormal if it contains a Hilbert basis for every cone spanned by a subset. We study such configurations from various perspectives, including triangulations, integer programming and Gröbner bases. Our main result is a bijection between virtual chambers of the configuration and virtual initial ideals of the associated… (More)

- Fabrizio Catanese, Serkan Hoşten, Amit Khetan, Bernd Sturmfels
- 2006

Maximum likelihood estimation in statistics leads to the problem of maximizing a product of powers of polynomials. We study the algebraic degree of the critical equations of this optimization problem. This degree is related to the number of bounded regions in the corresponding arrangement of hypersurfaces, and to the Euler characteristic of the complexified… (More)

- Serkan Hosten, Rekha R. Thomas
- Math. Program.
- 2003

The set of all group relaxations of an integer program contains certain special members called Gomory relaxations. A family of integer programs with a fixed coefficient matrix and cost vector but varying right hand sides is a Gomory family if every program in the family can be solved by one of its Gomory relaxations. In this paper, we characterize Gomory… (More)