Adam Kurpisz, Samuli Leppänen, Monaldo Mastrolilli
{"title":"Tight Sum-of-Squares lower bounds for binary polynomial optimization problems","authors":"Adam Kurpisz, Samuli Leppänen, Monaldo Mastrolilli","doi":"10.1145/3626106","DOIUrl":null,"url":null,"abstract":"For binary polynomial optimization problems of degree 2 d with n variables Sakaue, Takeda, Kim and Ito [SIAM J. Optim., 2017] proved that the \\(\\lceil \\frac{n+2d-1}{2}\\rceil \\) th semidefinite (SDP) relaxation in the SoS/Lasserre hierarchy of SDP relaxations provides the exact optimal value. When n is an odd number, we show that their analysis is tight, i.e. we prove that \\(\\frac{n+2d-1}{2} \\) levels of the SoS/Lasserre hierarchy are also necessary. Laurent [Math. Oper. Res., 2003] showed that the Sherali-Adams hierarchy requires n levels to detect the empty integer hull of a linear representation of a set with no integral points. She conjectured that the SoS/Lasserre rank for the same problem is n − 1. In this paper we disprove this conjecture and derive lower and upper bounds for the rank.","PeriodicalId":44045,"journal":{"name":"ACM Transactions on Computation Theory","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2023-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"16","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACM Transactions on Computation Theory","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1145/3626106","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
引用次数: 16
Abstract
For binary polynomial optimization problems of degree 2 d with n variables Sakaue, Takeda, Kim and Ito [SIAM J. Optim., 2017] proved that the \(\lceil \frac{n+2d-1}{2}\rceil \) th semidefinite (SDP) relaxation in the SoS/Lasserre hierarchy of SDP relaxations provides the exact optimal value. When n is an odd number, we show that their analysis is tight, i.e. we prove that \(\frac{n+2d-1}{2} \) levels of the SoS/Lasserre hierarchy are also necessary. Laurent [Math. Oper. Res., 2003] showed that the Sherali-Adams hierarchy requires n levels to detect the empty integer hull of a linear representation of a set with no integral points. She conjectured that the SoS/Lasserre rank for the same problem is n − 1. In this paper we disprove this conjecture and derive lower and upper bounds for the rank.