[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-127485-en":3,"doc-seo-127485-105":31,"detail-sidebar-cat-0-en-105":92},{"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":28,"seo_description":14,"update_tm":29,"read_time":30},127485,962085662650,"Jiven","https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0",8,"Research & Report","Hide and Seek - Scaling Machine Learning for Combinatorial Optimization via the Probabilistic Method","Applying deep learning to real-world hard combinatorial problems offers major promise, with Boolean SAT as a central target for both theory and practice. A key obstacle is that training data rely on random formulas far smaller than practical instances, since correctly labeling larger random formulas becomes intractable, and SAT-satisfiable cases are often too easy. The probabilistic method is used to generate correctly labeled random k-CNF formulas of arbitrary size without solving the decision problem. The generator tunes classification difficulty and enables training and evaluation of models on formulas with 10,000 variables, plus a novel prefix-computation classifier that outperforms random guessing.","arXiv :2211 . 15368v1 [ cs .AI] 21 Nov 2022  \nHide and Seek: Scaling Machine Learning for Combinatorial Optimization via the Probabilistic Method ∗  \nDimitris Achlioptas 1 , Amrit Daswaney2 , and Periklis A. Papakonstantinou3  \n1 University of Athens  \n2 Frogdata  \n3 Rutgers University  \nAbstract  \nApplying deep learning to solve real-life instances of hard combinatorial problems has tremendous potential. Research in this direction has focused on the Boolean satis-􀀌ability (SAT) problem, both because of its theoretical centrality and practical importance. A major roadblock faced, though, is that training sets are restricted to random formulas of size several orders of magnitude smaller than formulas of practical interest, raising serious concerns about generalization. This is because labeling random formulas of increasing size rapidly becomes intractable. By exploiting the probabilistic method in a fundamental way, we remove this roadblock entirely: we show how to generate correctly labeled random formulas of any desired size, without having to solve the underlying decision problem. Moreover, the di􀀎culty of the classi􀀌cation task for the formulas produced by our generator is tunable by varying a simple scalar parameter. This opens up an entirely new level of sophistication for the machine learning methods that can be brought to bear on Satis􀀌ability. Using our generator, we train existing state-of-the-art models for the task of predicting satis􀀌ability on formulas with 10,000 variables. We 􀀌nd that they do no better than random guessing. As a 􀀌rst indication of what can be achieved with the new generator, we present a novel classi􀀌er that performs signi􀀌cantly better than random guessing (99%) on the same datasets, formost di􀀎culty levels. Crucially, unlike past approaches that learn based on syntactic features of a formula, our classi􀀌er performs its learning on a short pre􀀌x of a solver's computation, an approach that we expect to be of independent interest.  \n∗ Authors are listed in alphabetical order. Corresponding author: Amrit Daswaney (amrit[daswaney@gmail.com](daswaney@gmail.com))  \n1 Introduction  \nMachine learning has been successfully applied to a wide range of domains where the instance distribution is not understood and thus no clear mathematical formulation of the problem can be constructed [8] . Recently, there has been great interest in investigating whether it can also be applied to combinatorial problems.  \nCentral in this e􀀋ort is Boolean Satis􀀌ability (SAT), the canonical NP-complete problem [13] . Besides its immense theoretical importance, SAT has several practical applications, ranging from hardware and software veri􀀌cation to planning and scheduling [19] . So far, researchers have built machine learning methods to predict satis􀀌ability [34, 10] and/or to 􀀌nda satisfying assignment [34, 4] .  \nA major roadblock in the e􀀋ort to use machine learning for satis􀀌ability is the absence of datasets comprising large labeled formulas from distributions for which the decision problem appears hard. Speci􀀌cally, in order to form a (balanced) training set, all existing works generate random 3-CNF formulas \\at the threshold,\" so that the probability of satis􀀌ability is (approximately) 1/2, and then use a complete (DPLL) solver to classify them, i.e., to decide whether each one is satis􀀌able or not. Since resolution proofs of unsatis􀀌ability (and thus DPLL solver executions) for random formulas are exponentially large in the number of variables [12], this means that they can only label relatively small formulas, i.e., orders of magnitude smaller than practically relevant formulas. To make things worse, unlike their unsatis􀀌able counterparts, the satis􀀌able formulas generated in this manner are very easy for modern SAT solvers.  \nThe use of formulas generated as above for training raises serious concerns about the potential generalization to formulas of practically relevant size. This is due not only to the very small ","cbCain0pu56gvAvs","https://ap.wps.com/l/cbCain0pu56gvAvs","pdf",1193826,2,1,18,"English","en",105,"# Abstract\n# Introduction\n## Machine learning for SAT and applications\n## Roadblocks: limited dataset size and generalization risk\n## Probabilistic-method generator for large labeled formulas\n## Overview of random k-CNF generation and SAT threshold","[{\"question\":\"Why is training data labeling a bottleneck for learning SAT?\",\"answer\":\"Labeling larger random SAT formulas becomes intractable, so existing datasets contain formulas orders of magnitude smaller than practical instances. Additionally, satisfiable formulas generated near the threshold can be very easy for modern solvers.\"},{\"question\":\"What does the proposed generator achieve using the probabilistic method?\",\"answer\":\"It efficiently generates arbitrarily large, correctly labeled random k-CNF formulas without having to solve the underlying decision problem. It also allows difficulty to be tuned from easy to extremely hard cases.\"},{\"question\":\"How does the new classifier differ from past approaches?\",\"answer\":\"Instead of learning from syntactic features of a formula, it learns from a short prefix of a solver’s computation. This design enables strong performance (reported up to 99%) on the same datasets for difficult levels.\"}]","Hide and Seek - Scaling Machine Learning for Combinatorial Optimization via the Probabilistic Method | PDF",1785939409,45,{"code":4,"msg":32,"data":33},"ok",{"site_id":25,"language":24,"slug":34,"title":13,"keywords":35,"description":14,"schema_data":36,"social_meta":87,"head_meta":89,"extra_data":91,"updated_unix":29},"hide-and-seek-scaling-machine-learning-for-combinatorial-optimization-via-the-probabilistic-method","",{"@graph":37,"@context":86},[38,54,69],{"@type":39,"itemListElement":40},"BreadcrumbList",[41,45,48,51],{"item":42,"name":43,"@type":44,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":46,"name":47,"@type":44,"position":20},"https://docshare.wps.com/document/","Document",{"item":49,"name":12,"@type":44,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":44,"position":53},"https://docshare.wps.com/document/hide-and-seek-scaling-machine-learning-for-combinatorial-optimization-via-the-probabilistic-method/127485/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":24,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":42,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-22","2026-08-05",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"Why is training data labeling a bottleneck for learning SAT?","Question",{"text":76,"@type":77},"Labeling larger random SAT formulas becomes intractable, so existing datasets contain formulas orders of magnitude smaller than practical instances. Additionally, satisfiable formulas generated near the threshold can be very easy for modern solvers.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"What does the proposed generator achieve using the probabilistic method?",{"text":81,"@type":77},"It efficiently generates arbitrarily large, correctly labeled random k-CNF formulas without having to solve the underlying decision problem. It also allows difficulty to be tuned from easy to extremely hard cases.",{"name":83,"@type":74,"acceptedAnswer":84},"How does the new classifier differ from past approaches?",{"text":85,"@type":77},"Instead of learning from syntactic features of a formula, it learns from a short prefix of a solver’s computation. This design enables strong performance (reported up to 99%) on the same datasets for difficult levels.","https://schema.org",{"og:url":52,"og:type":88,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":90,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":93},[94,98,102,106,111,116,121,124,129,132,136],{"id":21,"doc_module":4,"doc_module_name":47,"category_name":95,"show_sort_weight":96,"slug":97},"Story & Novel",90,"story-novel",{"id":20,"doc_module":4,"doc_module_name":47,"category_name":99,"show_sort_weight":100,"slug":101},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":47,"category_name":103,"show_sort_weight":104,"slug":105},"Exam",70,"exam",{"id":107,"doc_module":4,"doc_module_name":47,"category_name":108,"show_sort_weight":109,"slug":110},5,"Comic",60,"comic",{"id":112,"doc_module":4,"doc_module_name":47,"category_name":113,"show_sort_weight":114,"slug":115},6,"Technology",50,"technology",{"id":117,"doc_module":4,"doc_module_name":47,"category_name":118,"show_sort_weight":119,"slug":120},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":47,"category_name":12,"show_sort_weight":122,"slug":123},30,"research-report",{"id":125,"doc_module":4,"doc_module_name":47,"category_name":126,"show_sort_weight":127,"slug":128},9,"Religion & Spirituality",20,"religion-spirituality",{"id":127,"doc_module":4,"doc_module_name":47,"category_name":130,"show_sort_weight":127,"slug":131},"World Cup","world-cup",{"id":133,"doc_module":4,"doc_module_name":47,"category_name":134,"show_sort_weight":133,"slug":135},10,"Lifestyle","lifestyle",{"id":137,"doc_module":4,"doc_module_name":47,"category_name":138,"show_sort_weight":107,"slug":139},19,"General","general"]