[{"department":[{"_id":"64"}],"year":"2011","date_updated":"2019-01-03T13:13:55Z","title":"Solving the Closest Vector Problem with respect to Lp Norms","_id":"2987","date_created":"2018-06-05T07:49:32Z","abstract":[{"text":"In this paper, we present a deterministic algorithm for the closest vector\r\nproblem for all l_p-norms, 1 < p < \\infty, and all polyhedral norms, especially\r\nfor the l_1-norm and the l_{\\infty}-norm. We achieve our results by introducing\r\na new lattice problem, the lattice membership problem. We describe a\r\ndeterministic algorithm for the lattice membership problem, which is a\r\ngeneralization of Lenstra's algorithm for integer programming. We also describe\r\na polynomial time reduction from the closest vector problem to the lattice\r\nmembership problem. This approach leads to a deterministic algorithm that\r\nsolves the closest vector problem for all l_p-norms, 1 < p < \\infty, in time p\r\nlog_2 (r)^{O (1)} n^{(5/2+o(1))n} and for all polyhedral norms in time (s log_2\r\n(r))^{O (1)} n^{(2+o(1))n}, where s is the number of constraints defining the\r\npolytope and r is an upper bound on the coefficients used to describe the\r\nconvex body.","lang":"eng"}],"author":[{"first_name":"Johannes","last_name":"Blömer","id":"23","full_name":"Blömer, Johannes"},{"full_name":"Naewe, Stefanie","last_name":"Naewe","first_name":"Stefanie"}],"citation":{"ama":"Blömer J, Naewe S. Solving the Closest Vector Problem with respect to Lp Norms. *arXiv:11043720*. 2011.","mla":"Blömer, Johannes, and Stefanie Naewe. “Solving the Closest Vector Problem with Respect to Lp Norms.” *ArXiv:1104.3720*, 2011.","short":"J. Blömer, S. Naewe, ArXiv:1104.3720 (2011).","apa":"Blömer, J., & Naewe, S. (2011). Solving the Closest Vector Problem with respect to Lp Norms. *ArXiv:1104.3720*.","ieee":"J. Blömer and S. Naewe, “Solving the Closest Vector Problem with respect to Lp Norms,” *arXiv:1104.3720*. 2011.","bibtex":"@article{Blömer_Naewe_2011, title={Solving the Closest Vector Problem with respect to Lp Norms}, journal={arXiv:1104.3720}, author={Blömer, Johannes and Naewe, Stefanie}, year={2011} }","chicago":"Blömer, Johannes, and Stefanie Naewe. “Solving the Closest Vector Problem with Respect to Lp Norms.” *ArXiv:1104.3720*, 2011."},"user_id":"25078","publication":"arXiv:1104.3720","type":"preprint","status":"public"}]