{"title":"遗传差异与行列式下界的密切关系","authors":"Haotian Jiang, Victor Reis","doi":"10.1137/1.9781611977066.24","DOIUrl":null,"url":null,"abstract":"In seminal work, Lovász, Spencer, and Vesztergombi [European J. Combin., 1986] proved a lower bound for the hereditary discrepancy of a matrix A ∈ Rm×n in terms of the maximum |det(B)|1/k over all k × k submatrices B of A. We show algorithmically that this determinant lower bound can be off by at most a factor of O( √ log(m) · log(n)), improving over the previous bound of O(log(mn)· √ log(n)) given by Matoušek [Proc. of the AMS, 2013]. Our result immediately implies herdisc(F1 ∪ F2) ≤ O( √ log(m) · log(n)) · max(herdisc(F1), herdisc(F2)), for any two set systems F1,F2 over [n] satisfying |F1 ∪ F2| = m. Our bounds are tight up to constants when m = O(poly(n)) due to a construction of Pálvölgyi [Discrete Comput. Geom., 2010] or the counterexample to Beck’s three permutation conjecture by Newman, Neiman and Nikolov [FOCS, 2012]. University of Washington, Seattle, USA. jhtdavid@cs.washington.edu. University of Washington, Seattle, USA. voreis@cs.washington.edu.","PeriodicalId":93491,"journal":{"name":"Proceedings of the SIAM Symposium on Simplicity in Algorithms (SOSA)","volume":"61 1","pages":"308-313"},"PeriodicalIF":0.0000,"publicationDate":"2021-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":"{\"title\":\"A Tighter Relation Between Hereditary Discrepancy and Determinant Lower Bound\",\"authors\":\"Haotian Jiang, Victor Reis\",\"doi\":\"10.1137/1.9781611977066.24\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In seminal work, Lovász, Spencer, and Vesztergombi [European J. Combin., 1986] proved a lower bound for the hereditary discrepancy of a matrix A ∈ Rm×n in terms of the maximum |det(B)|1/k over all k × k submatrices B of A. We show algorithmically that this determinant lower bound can be off by at most a factor of O( √ log(m) · log(n)), improving over the previous bound of O(log(mn)· √ log(n)) given by Matoušek [Proc. of the AMS, 2013]. Our result immediately implies herdisc(F1 ∪ F2) ≤ O( √ log(m) · log(n)) · max(herdisc(F1), herdisc(F2)), for any two set systems F1,F2 over [n] satisfying |F1 ∪ F2| = m. Our bounds are tight up to constants when m = O(poly(n)) due to a construction of Pálvölgyi [Discrete Comput. Geom., 2010] or the counterexample to Beck’s three permutation conjecture by Newman, Neiman and Nikolov [FOCS, 2012]. University of Washington, Seattle, USA. jhtdavid@cs.washington.edu. University of Washington, Seattle, USA. voreis@cs.washington.edu.\",\"PeriodicalId\":93491,\"journal\":{\"name\":\"Proceedings of the SIAM Symposium on Simplicity in Algorithms (SOSA)\",\"volume\":\"61 1\",\"pages\":\"308-313\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2021-08-18\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"2\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Proceedings of the SIAM Symposium on Simplicity in Algorithms (SOSA)\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1137/1.9781611977066.24\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the SIAM Symposium on Simplicity in Algorithms (SOSA)","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1137/1.9781611977066.24","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
A Tighter Relation Between Hereditary Discrepancy and Determinant Lower Bound
In seminal work, Lovász, Spencer, and Vesztergombi [European J. Combin., 1986] proved a lower bound for the hereditary discrepancy of a matrix A ∈ Rm×n in terms of the maximum |det(B)|1/k over all k × k submatrices B of A. We show algorithmically that this determinant lower bound can be off by at most a factor of O( √ log(m) · log(n)), improving over the previous bound of O(log(mn)· √ log(n)) given by Matoušek [Proc. of the AMS, 2013]. Our result immediately implies herdisc(F1 ∪ F2) ≤ O( √ log(m) · log(n)) · max(herdisc(F1), herdisc(F2)), for any two set systems F1,F2 over [n] satisfying |F1 ∪ F2| = m. Our bounds are tight up to constants when m = O(poly(n)) due to a construction of Pálvölgyi [Discrete Comput. Geom., 2010] or the counterexample to Beck’s three permutation conjecture by Newman, Neiman and Nikolov [FOCS, 2012]. University of Washington, Seattle, USA. jhtdavid@cs.washington.edu. University of Washington, Seattle, USA. voreis@cs.washington.edu.