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A Safe Computational Framework for Integer Programming applied to Chvátal's Conjecture [article]

Leon Eifler, Ambros Gleixner, Jonad Pulaj
2020 arXiv   pre-print
We describe a general and safe computational framework that provides integer programming results with the degree of certainty that is required for machine-assisted proofs of mathematical theorems.  ...  The viability of this approach is tested by applying it to finite cases of Chv\'atal's conjecture, a long-standing open question in extremal combinatorics.  ...  The goal of this paper is to demonstrate how these tools can be employed to create a safe computational framework for investigating a particular mathematical application by integer programming.  ... 
arXiv:1809.01572v2 fatcat:6ryrbzcnzrhp3epuythuyfwaaq

Finding Cuts of Bounded Degree: Complexity, FPT and Exact Algorithms, and Kernelization

Guilherme C. M. Gomes, Ignasi Sau, Michael Wagner
2019 International Symposium on Parameterized and Exact Computation  
[IPEC 2018], is a polynomial kernel for d-Cut for every positive integer d, parameterized by the distance to a cluster graph.  ...  The degree bound in the hardness result is unlikely to be improved, as it would disprove a long-standing conjecture in the context of internal partitions.  ...  It is conjectured that, for every positive integer r, there exists some constant n r for which every r-regular graph with more than n r vertices has an internal partition [2, 10] (the conjecture for  ... 
doi:10.4230/lipics.ipec.2019.19 dblp:conf/iwpec/GomesS19 fatcat:tvr5pjyxlrckfbz64j3zdc3pdy

Finding cuts of bounded degree: complexity, FPT and exact algorithms, and kernelization [article]

Guilherme C. M. Gomes, Ignasi Sau
2019 arXiv   pre-print
[IPEC 2018], is a polynomial kernel for d-CUT for every positive integer d, parameterized by the distance to a cluster graph.  ...  The degree bound in the hardness result is unlikely to be improved, as it would disprove a long-standing conjecture in the context of internal partitions.  ...  It is conjectured that, for every positive integer r, there exists some constant n r for which every r-regular graph with more than n r vertices has an internal partition [2, 10] (the conjecture for  ... 
arXiv:1905.03134v1 fatcat:mi5454pvhrf2rblgrn7q3iv3de

Packing and Covering in Combinatorics

R. A. Cuninghame-Green, A. Schrijver
1980 Journal of the Operational Research Society  
certain integer linear programs.  ...  Thus attention is given to the Rogers bound for sphere-packings, Lloyd's theorem on perfect codes, Ramsey's theorem, graphtheoretical results of Konig, Menger, Turan and Tutte, Delsarte's linear programming  ...  In order to find an upper bound for A(8, 3) , we apply linear programming.  ... 
doi:10.2307/2581088 fatcat:szdtpgrkzrfvvehhovqz3r6vxq

Group Testing: An Information Theory Perspective

Matthew Aldridge, Oliver Johnson, Jonathan Scarlett
2019 Foundations and Trends in Communications and Information Theory  
We cover several related developments: efficient algorithms with practical storage and computation requirements, achievability bounds for optimal decoding methods, and algorithmindependent converse bounds  ...  Otherwise, pick half of the items in A, and call this set B. Perform a single test of the pool B. 4. If the test is positive, set A := B. If the test is negative, set A := A \ B. Return to Step 3.  ...  Acknowledgments We are very grateful to Sidharth Jaggi, who made significant contributions to this project in its early stages and has provided helpful advice and feedback on its later drafts.  ... 
doi:10.1561/0100000099 fatcat:t3paagtzy5clnjo3t3ikdhwfpq

Heeding the Call for Change Suggestions for Curricular Action THE MATHEMATICAL ASSOCIATION OF AMERICA MAA Notes and Reports Series

Lynn Steen
unpublished
One consequence is that some computer science programs might like a geometry course for their majors.  ...  Nowadays, most require some computer work (programming, computer use, or computer science) as well as some applied work (statistics, differential equations, etc.) among the electives.  ... 
fatcat:cq6twli67fgkjd3nnth2q3e4ii