[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-272866-105":3,"detail-sidebar-cat-0-en-105":81,"doc-detail-272866-en":130},{"code":4,"msg":5,"data":6},0,"ok",{"site_id":7,"language":8,"slug":9,"title":10,"keywords":11,"description":12,"schema_data":13,"social_meta":74,"head_meta":76,"extra_data":78,"updated_unix":80},105,"en","the-projection-games-conjecture-and-the-hardness-of-approximation-of-super-sat-and-related-problems","The Projection Games Conjecture and the Hardness of Approximation of super-SAT and related problems","","The Super-SAT (SSAT) problem was proposed to establish NP-hardness of approximating two lattice problems: the Shortest Vector Problem (SVP) and the Closest Vector Problem (CVP). The authors prove a conjecture that SSAT is NP-hard to approximate under a polylogarithmic gap, assuming the Projection Games Conjecture (PGC). They further reduce SSAT to the Nearest Codeword Problem (NCP) and the Learning Halfspace Problem (LHP), obtaining analogous inapproximability bounds. Under PGC, hardness extends to polynomial approximation factors.",{"@graph":14,"@context":73},[15,34,56],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/document/","Document",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/document/research-report/","Research & Report",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/document/the-projection-games-conjecture-and-the-hardness-of-approximation-of-super-sat-and-related-problems/272866/",4,{"url":32,"name":10,"@type":35,"image":36,"author":41,"headline":10,"publisher":44,"fileFormat":47,"inLanguage":8,"description":12,"dateModified":48,"datePublished":49,"encodingFormat":47,"isAccessibleForFree":50,"interactionStatistic":51},"DigitalDocument",{"url":37,"@type":38,"width":39,"height":40},"https://docshare.wps.com/thumbnails/the-projection-games-conjecture-and-the-hardness-of-approximation-of-super-sat-and-related-problems/272866.png","ImageObject",300,407,{"name":42,"@type":43},"Kurz","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-11-08","2026-09-15",true,{"@type":52,"interactionType":53,"userInteractionCount":55},"InteractionCounter",{"@type":54},"ViewAction",19,{"@type":57,"mainEntity":58},"FAQPage",[59,65,69],{"name":60,"@type":61,"acceptedAnswer":62},"What does the paper aim to prove about Super-SAT (SSAT)?","Question",{"text":63,"@type":64},"It proves a conjecture showing SSAT is NP-hard to approximate within stated factors, assuming the Projection Games Conjecture (PGC).","Answer",{"name":66,"@type":61,"acceptedAnswer":67},"How is SSAT related to lattice problems like SVP and CVP?",{"text":68,"@type":64},"The results imply hardness of approximation for SVP and CVP within polynomial factors when PGC holds, by establishing an approximation-preserving pathway from SSAT.",{"name":70,"@type":61,"acceptedAnswer":71},"Which additional problems are obtained via reductions from SSAT?",{"text":72,"@type":64},"The paper reduces SSAT to the Nearest Codeword Problem (NCP) and the Learning Halfspace Problem (LHP), yielding NP-hardness of approximation bounds that again improve under PGC.","https://schema.org",{"og:url":32,"og:type":75,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":77,"canonical":32},"index,follow",{"doc_id":79,"site_id":7},272866,1789468197,{"code":4,"msg":82,"data":83},"success",[84,88,92,96,101,106,111,115,120,123,127],{"id":22,"doc_module":4,"doc_module_name":25,"category_name":85,"show_sort_weight":86,"slug":87},"Story & Novel",90,"story-novel",{"id":26,"doc_module":4,"doc_module_name":25,"category_name":89,"show_sort_weight":90,"slug":91},"Literature",80,"literature",{"id":33,"doc_module":4,"doc_module_name":25,"category_name":93,"show_sort_weight":94,"slug":95},"Exam",70,"exam",{"id":97,"doc_module":4,"doc_module_name":25,"category_name":98,"show_sort_weight":99,"slug":100},5,"Comic",60,"comic",{"id":102,"doc_module":4,"doc_module_name":25,"category_name":103,"show_sort_weight":104,"slug":105},6,"Technology",50,"technology",{"id":107,"doc_module":4,"doc_module_name":25,"category_name":108,"show_sort_weight":109,"slug":110},7,"Healthcare",40,"healthcare",{"id":112,"doc_module":4,"doc_module_name":25,"category_name":29,"show_sort_weight":113,"slug":114},8,30,"research-report",{"id":116,"doc_module":4,"doc_module_name":25,"category_name":117,"show_sort_weight":118,"slug":119},9,"Religion & Spirituality",20,"religion-spirituality",{"id":118,"doc_module":4,"doc_module_name":25,"category_name":121,"show_sort_weight":118,"slug":122},"World Cup","world-cup",{"id":124,"doc_module":4,"doc_module_name":25,"category_name":125,"show_sort_weight":124,"slug":126},10,"Lifestyle","lifestyle",{"id":55,"doc_module":4,"doc_module_name":25,"category_name":128,"show_sort_weight":97,"slug":129},"General","general",{"code":4,"msg":82,"data":131},{"doc_id":79,"user_id":132,"nickname":42,"user_avatar":133,"doc_module":4,"category_id":112,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":55,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":139,"language":140,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":12,"update_tm":80,"read_time":144,"preview_page_count":4},2336478945635,"https://ap-avatar.wpscdn.com/davatar_6f874abed73319feea01a86fa6f0fab8","arXiv : 1907 .05548v 3 [ cs .CC] 5 Oct 2021  \nThe Projection Games Conjecture and the Hardness of Approximation of  \nsuper-SAT and related problems  \nPriyanka Mukhopadhyay 1,∗  \nInstitute for Quantum Computing & Department of Combinatorics and Optimization  \nUniversity of Waterloo  \n200 University Avenue West, Waterloo, ON N2L 3G1  \nAbstract  \nThe Super-SAT or SSAT problem was introduced by Dinur et al.[1, 2] to prove the NP-hardness of approximation of two popular lattice problems-Shortest Vector Problem(SVP) and Closest Vector Problem(CVP) . They conjectured that SSAT is NP-hard to approximate to within a factor of nc (c > 0 is constant), where n is the size of the SSAT instance. In this paper we prove this conjecture assuming the Projection Games Conjecture(PGC), given by Moshkovitz[3] . This implies hardness of approximation of SVP and CVP within polynomial factors, assuming PGC. We also reduce SSAT to the Nearest Codeword Problem(NCP) and Learning Halfspace Problem(LHP), as considered by Arora et al.[4] . This proves that both these problems are NP-hard to approximate within a factor of Nc′ / log log n(c′ > 0 is constant) where N is the size of the instances of the respective problems. Assuming PGC these problems are proved to be NP-hard to approximate within polynomial factors.  \nKeywords: Projection Games Conjecture, Inapproximability, Shortest Vector Problem, Closest Vector Problem, Learning Halfspace Problem, Nearest Codeword Problem  \n1. Introduction  \n1.1. SSAT and lattice problems  \nThe Super-SAT or SSAT problem was introduced by Dinur et al. [1] to prove the hardness of approximation of two popular lattice problems-the Shortest Vector Problem (SVP) and the Closest Vector Problem (CVP) . An n-dimensional lattice L is the set of integral linear combinations of n linearly independent vectorsin Rd , called the basis of the lattice. The goal of SVP is to 􀀌nd a shortest non-zero lattice vector. Given a  \n→−  \ntarget vector t , CVP aims at 􀀌nding a closest lattice vector to it.  \n􀀃 Corresponding author  \nEmail address: [mukhopadhyay.priyanka@gmail.com](mukhopadhyay.priyanka@gmail.com), [p3mukhop@uwaterloo.ca](p3mukhop@uwaterloo.ca) (Priyanka Mukhopadhyay)  \n1 Much of this work was done while the author was in Centre for Quantum Technologies, National University of Singapore.  \nPreprint submitted to Journal of Computer and System Sciences October 6, 2021  \nAlgorithms for these lattice problems are well-studied and have applications in factoring polynomials over rationals [5], integer programming [6, 7], cryptanalysis [8], checking the solvability by radicals [9], solving low-density subset-sum problems [10], cryptography [11, 12, 13, 14, 15, 16] .  \nThe SSAT problem is the gap version of SAT [F ], which is de􀀌ned as follows : An instance of SAT [F ] consists of a set of constraints or Boolean functions, called tests. The variables in each test take values from a 􀀌nite set F and each test has a set of satisfying assignments for its variables. The goal is to attach one assignment to each test such that consistency is maintained i.e. each variable gets the same value in all the tests in which it appears. The cardinality of the set of satisfying assignments is bounded by a polynomial in the number of variables.  \nIn SSAT, we attach integer weights to each assignment and call it a super-assignment. It is consistent if for each variable the sum of weights on each value is the same in all the tests with this variable. Values which get non-zero net weight are said to be simultaneously assigned to the variable. If each variable gets at least one value assigned we call it a non-trivial super-assignment. If for at least one test there exists at least one non-zero weighted assignment then we call it not-all-zero super-assignment. This also gives rise to the notion of norm of a super-assignment and accordingly two variants of SSAT has been de􀀌ned-the one in [1] for ℓ 1 norm and another by Dinur in [2] for ℓ∞ norm. An instance is accep","cbCait0teNb2DEuI","https://ap.wps.com/l/cbCait0teNb2DEuI","pdf",356542,28,"English","# Abstract\n# Introduction\n## SSAT and lattice problems\n## Label Cover (LC) and Projection Games Conjecture (PGC)","[{\"question\":\"What does the paper aim to prove about Super-SAT (SSAT)?\",\"answer\":\"It proves a conjecture showing SSAT is NP-hard to approximate within stated factors, assuming the Projection Games Conjecture (PGC).\"},{\"question\":\"How is SSAT related to lattice problems like SVP and CVP?\",\"answer\":\"The results imply hardness of approximation for SVP and CVP within polynomial factors when PGC holds, by establishing an approximation-preserving pathway from SSAT.\"},{\"question\":\"Which additional problems are obtained via reductions from SSAT?\",\"answer\":\"The paper reduces SSAT to the Nearest Codeword Problem (NCP) and the Learning Halfspace Problem (LHP), yielding NP-hardness of approximation bounds that again improve under PGC.\"}]","The Projection Games Conjecture and the Hardness of Approximation of super-SAT and related problems | PDF",71]