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Truncated Gröbner Bases for Integer Programming

R. R. Thomas, R. Weismantel
1997 Applicable Algebra in Engineering, Communication and Computing  
In this paper we introduce a multivariate grading of the toric ideal associated with the integer program min{cx : Ax = b, x ∈ IN n }, and a truncated Buchberger algorithm to solve the program.  ...  We thank Karin Gatermann and Bernd Sturmfels for helpful discussions and comments. This work was done while the first author was visiting ZIB.  ...  We refer to [2] and [5] for the theory of Gröbner bases and to [13] for toric ideals, their Gröbner bases and connections to integer programming and convex polytopes.  ... 
doi:10.1007/s002000050062 fatcat:epnfedbmzzcuxeaywiwla2pbea

Truncated Markov bases and Gröbner bases for Integer Programming [article]

Peter N. Malkin
2006 arXiv   pre-print
We present a new algorithm for computing a truncated Markov basis of a lattice. In general, this new algorithm is faster than existing methods.  ...  We also present a novel Groebner basis approach to solve a particular integer linear program as opposed to previous Groebner basis methods that effectively solved many different integer linear programs  ...  Acknowledgments I would like to thank Raymond Hemmecke and Laurence Wolsey for many fruitful discussions and helpful comments.  ... 
arXiv:math/0612615v1 fatcat:s6b677es5jgy7mgryorbzzorse

Truncated Groebner fans and lattice ideals [article]

Niels Lauritzen
2005 arXiv   pre-print
This "truncated" Groebner fan is usually much smaller than the full Groebner fan and offers the natural framework for conversion between truncated Groebner bases.  ...  Computational experience with the special Aardal-Lenstra integer programming knapsack problems is reported.  ...  Truncation of homogeneous ideals have proved very valuable in algebraic computations related to integer programming (see for example [7] ).  ... 
arXiv:math/0509247v1 fatcat:bvvf3jndxzhpfli23dxgjscbiu

On the Complexity of Toric Ideals [article]

Diego Cifuentes, Shmuel Onn
2019 arXiv   pre-print
This test leads to an efficient procedure for computing the reduced Gr\"obner basis. Similar results hold for Graver bases computation.  ...  We investigate the computational complexity of problems on toric ideals such as normal forms, Gr\"obner bases, and Graver bases.  ...  Both authors thank Bernd Sturmfels and the Max-Planck Institute in Leipzig for their hospitality.  ... 
arXiv:1902.01484v1 fatcat:6gjzadnahzc63bb22becv4zi3q

Computing generating sets of lattice ideals and Markov bases of lattices

Raymond Hemmecke, Peter N. Malkin
2009 Journal of symbolic computation  
Two areas of application for generating sets of lattice ideals and Markov bases lattices are algebraic statistics and integer programming.  ...  In this article, we present an algorithm for computing generating sets of lattice ideals or equivalently for computing Markov bases of lattices.  ...  Markov bases of lattices also have an application in integer programming.  ... 
doi:10.1016/j.jsc.2009.04.006 fatcat:raxeen3bobdh5e3br7nelnxw5m

The generic Gröbner walk

K. Fukuda, A.N. Jensen, N. Lauritzen, R. Thomas
2007 Journal of symbolic computation  
The Gröbner walk is an algorithm for conversion between Gröbner bases for different term orders.  ...  It is based on the polyhedral geometry of the Gröbner fan and involves tracking a line between cones representing the initial and target term order.  ...  Acknowledgements We are grateful to Peter Malkin for pointing out inaccuracies in our section on the classical Gröbner walk.  ... 
doi:10.1016/j.jsc.2006.09.004 fatcat:atbdidoufndc5pm76qnnlp7gqi

Partial Gröbner Bases for Multiobjective Integer Linear Optimization

Víctor Blanco, Justo Puerto
2009 SIAM Journal on Discrete Mathematics  
Later, Thomas and Weismantel [43] improved the Buchberger algorithm in its application to solve integer programs introducing truncated Gröbner bases.  ...  One of the outcomes of Gröbner bases theory was its application to integer programming, first published by Conti and Traverso [10].  ...  In particular, one can expect improvements in the efficiency of our algorithm based on the special structure of the integer program (see, for instance, Remark 3.1).  ... 
doi:10.1137/070698051 fatcat:jysiupewhfbgffmgm7nfmgk4wu

The generic Groebner walk [article]

K. Fukuda, A. N. Jensen, N. Lauritzen, R. Thomas
2005 arXiv   pre-print
The Groebner walk is an algorithm for conversion between Groebner bases for different term orders.  ...  We report on computations with toric ideals, where a version of our algorithm in certain cases computes test sets for hard integer knapsack problems significantly faster than the Buchberger algorithm.  ...  Now we may apply standard algebraic techniques in integer programming (cf.  ... 
arXiv:math/0501345v3 fatcat:s7pdwjek4jcnhcgkk436b6mv3e

Finding linear building-blocks for RTL synthesis of polynomial datapaths with fixed-size bit-vectors

Sivaram Gopalakrishnan, Priyank Kalla, M. Brandon Meredith, Florian Enescu
2007 Computer-Aided Design (ICCAD), IEEE International Conference on  
Given a polynomial, we identify a specific set of linear expressions and compute the Gröbner bases of their ideal (over non-UFD Z2m ) using syzygies.  ...  This basis serves as good building-blocks for the given computation. A decomposition is identified by subsequent Gröbner basis reduction.  ...  David Cox of Amherst College for informing us about the Gröbner bases computation over a non-UFD Z2m using the Möller's algorithm [10] .  ... 
doi:10.1109/iccad.2007.4397257 dblp:conf/iccad/GopalakrishnanKME07 fatcat:rro6phigt5awvhla5jzpyb7o3e

Using carry-truncated addition to analyze add-rotate-xor hash algorithms

Rebecca E. Field, Brant C. Jones
2013 Journal of Mathematical Cryptology  
We use truncated addition to analyze hash functions that are built from the bit operations add, rotate, and xor, such as Blake, Skein, and Cubehash.  ...  We introduce a truncated addition operation on pairs of N -bit binary numbers that interpolates between ordinary addition mod 2 N and bitwise addition in (Z/2Z) N .  ...  ACKNOWLEDGMENTS We thank Elizabeth Arnold for sharing her expertise on Gröbner basis algorithms and Nicky Mouha for helpful comments on an earlier draft of this work.  ... 
doi:10.1515/jmc-2012-0019 fatcat:bzly4y5vvvapboio5wp5a47ljm

Using carry-truncated addition to analyze add-rotate-xor hash algorithms [article]

Rebecca E. Field, Brant C. Jones
2013 arXiv   pre-print
We use truncated addition to analyze hash functions that are built from the bit operations add, rotate, and xor, such as Blake, Skein, and Cubehash.  ...  We introduce a truncated addition operation on pairs of N-bit binary numbers that interpolates between ordinary addition mod 2^N and bitwise addition in (Z/2Z)^N.  ...  ACKNOWLEDGMENTS We thank Elizabeth Arnold for sharing her expertise on Gröbner basis algorithms and Nicky Mouha for helpful comments on an earlier draft of this work.  ... 
arXiv:1303.4448v1 fatcat:qidhoesz2rcbrblduig2sycyn4

Partial Gröbner bases for multiobjective integer linear optimization [article]

Victor Blanco, Justo Puerto
2008 arXiv   pre-print
In this paper we present a new methodology for solving multiobjective integer linear programs using tools from algebraic geometry.  ...  It allows us to prove that this new construction is a test family for a family of multiobjective programs.  ...  A different truncation strategy may be based on the number of steps required to obtain the p-Gröbner basis.  ... 
arXiv:0709.1660v2 fatcat:grnakkrdyrdjnfc6fxppqi7wla

More Efficient Identifiability Verification in ODE Models by Reducing Non-Identifiability [article]

Ilia Ilmer, Alexey Ovchinnikov, Gleb Pogudin, Pedro Soto
2022 arXiv   pre-print
If a given model has parameters for which there may be infinitely many values, such parameters are called non-identifiable.  ...  We present a procedure for accelerating a global identifiability query by eliminating algebraically independent non-identifiable parameters.  ...  For an overview of F5-based solutions, see [7] . Parameter identifiability Solutions for the identifiability problem have implementations in various programming languages.  ... 
arXiv:2204.01623v1 fatcat:p5t2p5tmnrgmdmsghi7ebvmw6m

Flat modules and Gröbner bases over truncated discrete valuation rings [article]

Toshiro Hiranouchi, Yuichiro Taguchi
2009 arXiv   pre-print
We present basic properties of Gröbner bases of submodules of a free module of finite rank over a polynomial ring R with coefficients in a graded truncated discrete valuations ring A.  ...  As an application, we give a criterion for a finitely generated R-module to be flat over A. Its non-graded version is also given.  ...  We thank the organizers, Jaebum Sohn and Hisao Taya, of the 11th Japan-Korea Joint Seminar at Sendai for inviting us to the Seminar and providing us with the opportunity to write up the results in this  ... 
arXiv:0901.4819v2 fatcat:dftik4jhjfgnhj2wfrsjbgtsbq

The "Seven Dwarfs" of Symbolic Computation [chapter]

Erich L. Kaltofen
2011 Texts & Monographs in Symbolic Computation  
Exact linear algebra, integer lattices SymDwf 2. Exact polynomial and differential algebra, Gröbner bases SymDwf 3. Inverse symbolic problems, e.g., interpolation and parameterization SymDwf 4.  ...  Acknowledgments: I thank Bruno Salvy for his thoughtful comments.  ...  The first are the numerical optimization algorithms for semidefinite programming.  ... 
doi:10.1007/978-3-7091-0794-2_5 fatcat:y6fjj6ghrzek5ps4vig7uv5vpy
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