{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article17254","name":"The Biased Homogeneous r-Lin Problem","abstract":"The p-biased Homogeneous r-Lin problem (Hom-r-Lin_p) is the following: given a homogeneous system of r-variable equations over m{F}\u2082, the goal is to find an assignment of relative weight p that satisfies the maximum number of equations. In a celebrated work, H\u00e5stad (JACM 2001) showed that the unconstrained variant of this i.e., Max-3-Lin, is hard to approximate beyond a factor of 1\/2. This is also tight due to the naive random guessing algorithm which sets every variable uniformly from {0,1}. Subsequently, Holmerin and Khot (STOC 2004) showed that the same holds for the balanced Hom-r-Lin problem as well. In this work, we explore the approximability of the Hom-r-Lin_p problem beyond the balanced setting (i.e., p \u2260 1\/2), and investigate whether the (p-biased) random guessing algorithm is optimal for every p. Our results include the following: \r\n- The Hom-r-Lin_p problem has no efficient 1\/2 + 1\/2 (1 - 2p)^{r-2} + \u03b5-approximation algorithm for every p if r is even, and for p \u2208 (0,1\/2] if r is odd, unless NP \u2282 \u222a_{\u03b5>0}DTIME(2^{n^\u03b5}). \r\n- For any r and any p, there exists an efficient 1\/2 (1 - e^{-2})-approximation algorithm for Hom-r-Lin_p. We show that this is also tight for odd values of r (up to o_r(1)-additive factors) assuming the Unique Games Conjecture. Our results imply that when r is even, then for large values of r, random guessing is near optimal for every p. On the other hand, when r is odd, our results illustrate an interesting contrast between the regimes p \u2208 (0,1\/2) (where random guessing is near optimal) and p \u2192 1 (where random guessing is far from optimal). A key technical contribution of our work is a generalization of H\u00e5stad\u2019s 3-query dictatorship test to the p-biased setting.","keywords":["Biased Approximation Resistance","Constraint Satisfaction Problems"],"author":{"@type":"Person","name":"Ghoshal, Suprovat","givenName":"Suprovat","familyName":"Ghoshal","email":"mailto:suprovat.ghoshal@gmail.com","affiliation":"University of Michigan, Ann Arbor, MI, USA"},"position":47,"pageStart":"47:1","pageEnd":"47:14","dateCreated":"2022-09-15","datePublished":"2022-09-15","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":{"@type":"Person","name":"Ghoshal, Suprovat","givenName":"Suprovat","familyName":"Ghoshal","email":"mailto:suprovat.ghoshal@gmail.com","affiliation":"University of Michigan, Ann Arbor, MI, USA"},"copyrightYear":"2022","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.APPROX\/RANDOM.2022.47","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","citation":["https:\/\/doi.org\/10.1145\/3519935.3520072","https:\/\/doi.org\/10.1007\/s00037-008-0256-y","http:\/\/arxiv.org\/abs\/1310.1493"],"isPartOf":{"@type":"PublicationVolume","@id":"#volume6448","volumeNumber":245,"name":"Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX\/RANDOM 2022)","dateCreated":"2022-09-15","datePublished":"2022-09-15","editor":[{"@type":"Person","name":"Chakrabarti, Amit","givenName":"Amit","familyName":"Chakrabarti","email":"mailto:amit.chakrabarti@dartmouth.edu","sameAs":"https:\/\/orcid.org\/0000-0003-3633-9180","affiliation":"Dartmouth College, Hanover, NH, USA"},{"@type":"Person","name":"Swamy, Chaitanya","givenName":"Chaitanya","familyName":"Swamy","email":"mailto:cswamy@uwaterloo.ca","sameAs":"https:\/\/orcid.org\/0000-0003-1108-7941","affiliation":"University of Waterloo, Canada"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article17254","isPartOf":{"@type":"Periodical","@id":"#series116","name":"Leibniz International Proceedings in Informatics","issn":"1868-8969","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume6448"}}}