4ti2 -- A software package for algebraic, geometric and combinatorial problems on linear spaces

Theory of main algorithms implemented

Algebraic and Geometric Ideas in the Theory
of Discrete Optimization

graver
- R. Hemmecke. On the Positive Sum Property and the Computation of Graver test sets. Mathematical Programming, 96(2):247--269.
- R. Hemmecke. Exploiting Symmetries in the Computation of Graver Bases. e-print arXiv:math.CO/0410334, 2004.
- Chapter 3 in: J. A. De Loera, R. Hemmecke, M. Köppe. Algebraic and Geometric Ideas in the Theory of Discrete Optimization, MOS-SIAM Series on Optimization, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2013, xx + 313 pages.
groebner / markov
- A.M. Bigatti and R. LaScala and L. Robbiano. Computing toric ideals. Journal of Symbolic Computation 27 (1999), 351--365.
- R. Gebauer and H. M. Möller. On an installation of Buchberger's algorithm. Journal of Symbolic Computation 6 (1988), 275--286.
- R. Hemmecke and P. Malkin. Computing generating sets of lattice ideals. e-print arXiv:math.CO/0508359, 2005.
- S. Hosten and B. Sturmfels. GRIN: An implementation of Gröbner bases for integer programming. In: "Integer programming and combinatorial optimisation", E. Balas and J. Clausen, eds., LNCS 920, Springer-Verlag, 1995, 267--276.
- P. Malkin. Truncated Markov bases and Gröbner bases for Integer Programming. e-print arXiv:math.OC/0612615, 2006.
- Chapter 11 in: J. A. De Loera, R. Hemmecke, M. Köppe. Algebraic and Geometric Ideas in the Theory of Discrete Optimization, MOS-SIAM Series on Optimization, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2013, xx + 313 pages.
hilbert
- R. Hemmecke. On the Computation of Hilbert Bases of Cones. in: "Mathematical Software, ICMS 2002", A. M. Cohen, X.-S. Gao, N. Takayama, eds., World Scientific, 2002.
- Chapter 3 in: J. A. De Loera, R. Hemmecke, M. Köppe. Algebraic and Geometric Ideas in the Theory of Discrete Optimization, MOS-SIAM Series on Optimization, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2013, xx + 313 pages.
ppi
- M. Köppe. Erzeugende Mengen für gemischt-ganzzahlige Programme. Diploma thesis, Otto-von-Guericke-Universität Magdeburg, 1999. available from URL http://www.math.ucdavis.edu/~mkoeppe/art/mkoeppe-diplom.ps.
- U.-U. Haus, M. Köppe, and R. Weismantel. A primal all-integer algorithm based on irreducible solutions. Math. Programming, Series B, 96(2):205-246, 2003.