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+# Graver bases
+
+Project in mathematics developed at the [Chair of Discrete Optimization](https://www.epfl.ch/labs/disopt/) at the **École Polytechnique Fédérale de Lausanne**, Switzerland.
+
+* **Project director:** [Friedrich Eisenbrand](https://en.wikipedia.org/wiki/Friedrich_Eisenbrand)
+* **Project supervisor:** [Jana Cslovjecsek](https://www.epfl.ch/labs/disopt/page-10520-en-html/page-31595-en-html/jana-cslovjecsek/)  
+
+The final result of the project can be accessed here: **[Graver_bases.pdf](Graver_bases.pdf)**
+
+## Abstract
+
+In the project we study the latest techniques based on the **Graver bases** and its bounds as well as their application to the **N-Fold IP**, 
+a restricted formulation of the IP which has won relevance in the last decades given its theoretical properties and its wide applications. For 
+the **N-Fold IP** we introduce the best algorithm known for its resolution running in a roughly linear time as well as a quadratic augmentation 
+algorithm based on the properties of the Graver basis of the N-Fold matrix.
+
+The following presentations may be useful as an overview:
+
+* [Graver bases bounds augmentation algorithm](Graver_bases_presentation_mid_semester.pdf)
+* [N-Fold IP via LP rounding](Graver_bases_presentation_end_semester.pdf)
+
+
+## Bibliography
+
+* Graver, [On the foundations of linear and integer linear programming I](https://link.springer.com/article/10.1007/BF01681344), 1975.
+* Sturmfels, [Algebraic Recipes for Integer Programming](https://arxiv.org/abs/math/0310194), 2003.
+* Finhold, Hemmecke, [Lower bounds on the Graver complexity of M-fold matrices](https://arxiv.org/abs/1311.3853), 2013.
+* Eisenbrand, Hunkenschröder, Klein,[Faster Algorithms for Integer Programs with Block Structure](https://arxiv.org/abs/1802.06289), 2018.
+* Cslovjecsek, Eisenbrand, Weismantel, [N-fold integer programming via LP rounding](https://arxiv.org/abs/2002.07745), 2020.
+* Hemmecke, Onn, Romanchuk, [N-fold integer programming in cubic time](https://arxiv.org/pdf/1101.3267.pdf), 2011.
+* Eisenbrand, Weismantel, [Proximity results and faster algorithms for Integer Programming using the Steinitz Lemma](https://arxiv.org/abs/1707.00481), 2019.
+* De Loera, Hemmecke, Onn, Weismantel, [N-Fold integer programming](https://www.math.ucdavis.edu/~deloera/researchsummary/n-fold.pdf), 2006.
+* Hemmecke, [Exploiting Symmetries in the Computation of Graver Bases](https://www.researchgate.net/publication/2115495_Exploiting_Symmetries_in_the_Computation_of_Graver_Bases), 2004.
+* Hemmecke, Onn, Weismantel, [A polynomial oracle-time algorithm for convex integer minimization](https://link.springer.com/article/10.1007/s10107-009-0276-7), 2009.
+* Onn, [Convex discrete optimization](https://arxiv.org/pdf/math/0703575.pdf), 2007.
+* Onn, [Nonlinear discrete optimization](http://pi.math.cornell.edu/event/conf/billera65/notes/onn.pdf), 2010.
+* De Loera, Onn, [All linear and integer programs are slim 3-way transportation programs](https://www.math.ucdavis.edu/~deloera/researchsummary/universalitytransportation.pdf), 2006.
+* Steinitz, [Bedingt konvergente reihen und konvexe systeme](https://eudml.org/doc/149403), 1913.
+