[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86005-en":3,"doc-seo-86005-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},86005,1099514067415,"Rowan","https://ap-avatar.wpscdn.com/avatar/100002539d78ffe74a7?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779092875211072502",8,"Research & Report","A Better Analysis For PPSZ For 3-SAT","Revisiting Scheder’s analysis of the original PPSZ algorithm for 3-SAT, the work keeps the regular and irregular estimates unchanged while changing only their final recombination. The recombination is performed via an explicit linear-programming dual certificate expressed in common structural coordinates, with the numerical inequalities verified by exact rational interval computation. The improved Unique-3-SAT bonus yields a strictly better worst-case randomized running-time bound for general 3-SAT using the existing Scheder–Steinberger lifting theorem, without modifying either.","arXiv :2607 . 10697v 1 [ cs .DS] 12 Jul 2026  \nA Better Analysis For PPSZ For 3-SAT  \nTao Jiang Shaowei Cai  \nJuly 2026  \nAbstract  \nWe revisit Scheder’s analysis of the original PPSZ algorithm. Keeping his regular and irregular estimates unchanged, we express them in common structural coordinates and replace only their final recombination by an explicit linear-programming dual certificate. The old and new running-time bounds are  \n\n|  | Unique-3-SAT | general 3-SAT |\n| --- | --- | --- |\n| Scheder’s analysis | O ∗ (1 .306972377n ) | O ∗ (1 .307031594n ) |\n| this work | O ∗ (1 .306969598n ) | O ∗ (1 .307031578n ) . |\n\nIn both rows, the general-case bound is obtained by applying the same existing Scheder– Steinberger unique-to-general lifting theorem to the corresponding Unique-3-SAT analysis. To the best of our knowledge, O ∗ (1 .307031578n ) is the best currently known worst-case randomized running-time bound for general 3-SAT. Neither PPSZ nor the lifting theorem is modified. The numerical inequalities are certified by exact rational interval computation.  \n1 Introduction  \nThe PPSZ algorithm of Paturi, Pudl´ak, Saks, and Zane [2] processes the variables of a satisfiable CNF formula in a uniformly random order. At each variable it applies a bounded implication rule; if the value is not inferred, it guesses an unbiased bit. For Unique-3-SAT, the classical exponent is  \nP [PPSZ(F) = α] ≥ 2 −p0n−o(n), p0 = 2ln2 − 1 ,  \nso the corresponding running-time base is 2p0 = 1 .3070319 ... .  \nScheder [3, 4 , 5] obtained a stronger bound for the same algorithm. His full k = 3 argument derives two lower bounds, called the regular and irregular estimates. In the notation reconciled in Section 2, the final simplification in Section 6 of the full version is  \n | H |   n   | J1 | + 2|J0 |  \ngainR ≥ 10118 − 41391 , gain I ≥ 1380 . (1)  \nWriting irr = (|J1 | + 2|J0 |)/n and using |H|/n ≥ 1 − irr gives  \n1n max{gainR , gain I} ≥ max 􀀚 11−~~ ~~irr0118 − 411391 , 1irr380 􀀛 ≥ 151218 . (2)  \nThus Scheder’s published unique-case bonus is  \n 1   \nγold = = 0 .000065711657247995 . . . ,  \n15218  \nwith unrounded base 1 .306972376565153 ... .  \nOur argument starts from these two estimates. We retain a positive regular coefficient that is discarded in the simplification leading to (2), express both estimates in the common coordinates  \ni0 = |~~ ~~IDn0~~ ~~|~~ ~~ , i 1 = |~~ ~~IDn1~~ ~~|~~ ~~ , τ = |~~ ~~TwonCC~~ ~~|~~ ~~ ,  \nand combine them by a feasible dual solution of a three-variable linear program. For fixed numerical parameters, the two bounds have the form  \nLreg = A − Preg − 2Ai0 − Ai 1 + Sτ, Lirr = b0 i0 + b1 i 1 + bT τ .  \nFor the parameters fixed below, S > 0 and bT \u003C 0. Taking λ = b 1 /A gives  \nb0 − 2λA > 0, b1 − λA = 0, bT + λS > 0 ,  \nso the weighted average (λLreg + Lirr)/(1 + λ) has no negative structural coefficient. Theorem 1.1 (Unique-3-SAT) . Let  \nγnew = 0 .0000687793.  \nThere exist a finite implication strength w0 and an integer n0 such that, for every w ≥ w0 , every n ≥ n0 , and every 3-CNF formula F on n variables with unique satisfying assignment α ,  \nP [PPSZw(F) = α] ≥ 2 −p0n+γnewn = 2−n+s3n+γnewn.  \nConsequently, independent repetition gives a randomized algorithm with running time  \nO∗ (1 .306969598n ) .  \nHere PPSZw denotes standard uniform-order, unbiased-guessing PPSZ with implication strength w; Section 2 . 1 explains the equivalent bounded-width implementation. The fixed-parameter certificate gives  \nγ∗ = 0 .000068779380458836 . . . ,  \nand the theorem uses the strictly smaller decimal γ new. In particular,  \n0.0000687793 > 0.000065711657247995 . . . ,  \nso Theorem 1.1 strictly improves Scheder’s unique-case exponent.  \nApplying the existing unique-to-general theorem of Scheder and Steinberger [6] to the new unique-case bonus gives the following corollary. The lifting theorem is used without modification; only its numerical instantiation changes.  \nCorollary 1.2 (General 3-SAT via Scheder–Steinberger) . There exist a fin","cbCaibTrMxtP56kO","https://ap.wps.com/l/cbCaibTrMxtP56kO","pdf",335705,2,1,15,"English","en",105,"# Abstract\n# Introduction\n# Imported estimates and finite-strength conventions\n## Algorithmic convention and order of limits","[{\"question\":\"What aspect of Scheder’s PPSZ analysis is changed in this work?\",\"answer\":\"The regular and irregular estimates are kept, but their final recombination is replaced using an explicit linear-programming dual certificate in common structural coordinates.\"},{\"question\":\"Does the approach modify the PPSZ algorithm or the lifting theorem?\",\"answer\":\"No. Neither PPSZ nor the Scheder–Steinberger unique-to-general lifting theorem is modified; only the unique-case exponent used in the lifting changes.\"},{\"question\":\"How does the new result affect the worst-case randomized running time for general 3-SAT?\",\"answer\":\"The improved Unique-3-SAT exponent produces a strictly better lifted general-3-SAT bonus, giving a smaller running-time base for general 3-SAT.\"}]",1784207724,38,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"a-better-analysis-for-ppsz-for-3-sat","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,47,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":20},"https://docshare.wps.com/document/","Document",{"item":48,"name":12,"@type":43,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/a-better-analysis-for-ppsz-for-3-sat/86005/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-24","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What aspect of Scheder’s PPSZ analysis is changed in this work?","Question",{"text":75,"@type":76},"The regular and irregular estimates are kept, but their final recombination is replaced using an explicit linear-programming dual certificate in common structural coordinates.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Does the approach modify the PPSZ algorithm or the lifting theorem?",{"text":80,"@type":76},"No. Neither PPSZ nor the Scheder–Steinberger unique-to-general lifting theorem is modified; only the unique-case exponent used in the lifting changes.",{"name":82,"@type":73,"acceptedAnswer":83},"How does the new result affect the worst-case randomized running time for general 3-SAT?",{"text":84,"@type":76},"The improved Unique-3-SAT exponent produces a strictly better lifted general-3-SAT bonus, giving a smaller running-time base for general 3-SAT.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]