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The literature contains many encodings for even simple constraint types, with widely different performance, making encoding selection non-trivial. This work uses supervised machine learning to select effective encodings for pseudo-Boolean and linear integer constraints. It achieves strong results for unseen problem classes, analyzes feature importance, and compares multiple learning-method variants against AutoFolio under the same feature set.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/selecting-sat-encodings-constraints/504966/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/selecting-sat-encodings-constraints/504966.png","ImageObject",300,407,{"name":92,"@type":93},"asda1da","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-21","2026-10-04",true,{"@type":102,"interactionType":103,"userInteractionCount":39},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What problem does the paper address?","Question",{"text":112,"@type":113},"Selecting suitable SAT encodings for pseudo-Boolean and linear integer constraint instances when many alternative encodings exist with varying performance.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How is encoding selection performed?",{"text":117,"@type":113},"It uses a supervised machine learning approach trained on a corpus of 614 instances across 49 problem classes.",{"name":119,"@type":110,"acceptedAnswer":120},"Why are unseen problem classes important in this work?",{"text":121,"@type":113},"The study evaluates two split settings and argues split-by-class is realistic, testing robustness when test instances belong to classes not seen during training.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},504966,1791287048,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":39,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":139,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":144,"read_time":145},3985747861852,"https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d","Springer Nature 2021 LATEX template  \nLearning to Select SAT Encodings for Pseudo-Boolean and Linear Integer Constraints  \nFelix Ulrich-Oltean, Peter Nightingale and James Alfred  \nWalker  \nDepartment of Computer Science, University of York, Deramore Lane, York, YO10 5GH, United Kingdom.  \n*Corresponding author(s) . E-mail(s): [felix.ulrich-oltean@york.ac.uk](felix.ulrich-oltean@york.ac.uk) ; Contributing authors: [peter.nightingale@york.ac.uk](peter.nightingale@york.ac.uk) ;  \n[james.walker@york.ac.uk](james.walker@york.ac.uk) ;  \nAbstract  \nMany constraint satisfaction and optimisation problems can be solved effectively by encoding them as instances of the Boolean Satisfiability problem (SAT) . However, even the simplest types of constraints have many encodings in the literature with widely varying performance, and the problem of selecting suitable encodings for a given problem instance is not trivial. We explore the problem of selecting encodings for pseudo-Boolean and linear constraints using a supervised machine learning approach. We show that it is possible to select encodings effectively using a standard set of features for constraint problems; however we obtain better performance with a new set of features specifically designed for the pseudo-Boolean and linear constraints. In fact, we achieve good results when selecting encodings for unseen problem classes. Our results compare favourably to AutoFolio when using the same feature set. We discuss the relative importance of instance features to the task of selecting the best encodings, and compare several variations of the machine learning method.  \nKeywords: constraint programming, SAT encodings, machine learning, global constraints, pseudo-Boolean constraints, linear constraints  \nSpringer Nature 2021 LATEX template  \n2 Learning to Select SAT Encodings  \n1 Introduction  \nMany constraint satisfaction and optimisation problems can be solved effectively by encoding them as instances of the Boolean Satisfiability problem (SAT) . Modern SAT solvers are remarkably effective even with large formulas, and have proven to be competitive with (and often faster than) CP solvers (including those with conflict learning) . However, even the simplest types of constraints have many encodings in the literature with widely varying performance, and the problem of predicting suitable encodings is not trivial.  \nWe explore the problem of selecting encodings for constraints of the form P qi ei ⋄ k where ⋄ ∈ {\u003C, ≤ , = , , ≥,>}, q1 . . . qn are integer coefficients, k isan integer constant and ei are decision variables or simple expressions containing a single decision variable (such as negation) . We separate these constraints into two classes: pseudo-Boolean (PB) when all ei are Boolean; and linear integer (LI) when there exists an integer expression ei. We treat these two classes separately, selecting one encoding for each class when encoding an instance.  \nWe select from a set of state-of-the-art encodings, including all eight encodings of Bofill et al [1–3] which are extensions of the Generalized Totalizer [4], Binary Decision Diagram [5], Global Polynomial Watchdog [6], Local Polynomial Watchdog [6], Sequential Weight Counter [7], and n-Level Modulo Totalizer [8] . All eight of these encodings are for pseudo-Boolean constraints combined with at-most-one (AMO) sets of terms (where at most one of the corresponding ei Boolean expressions are true in a solution) . The AMO sets come from an integer variable or are detected automatically [9] as described in Section 2.1. We also use an encoding named Tree which is described in this paper.  \nThe context for this work is Savile Row [10], a constraint modelling tool that takes the modelling language Essence Prime and can produce output for various types of solver, including CP, SAT, and recently SMT [11] . When encoding a constraint to SAT, two different approaches may be taken depending on the type of constraint. Some constraint types are decompos","cbCaiikzHJfz3UaF","https://ap.wps.com/l/cbCaiikzHJfz3UaF","pdf",1250156,36,"English","# Abstract\n# Introduction\n## Contributions","[{\"question\":\"What problem does the paper address?\",\"answer\":\"Selecting suitable SAT encodings for pseudo-Boolean and linear integer constraint instances when many alternative encodings exist with varying performance.\"},{\"question\":\"How is encoding selection performed?\",\"answer\":\"It uses a supervised machine learning approach trained on a corpus of 614 instances across 49 problem classes.\"},{\"question\":\"Why are unseen problem classes important in this work?\",\"answer\":\"The study evaluates two split settings and argues split-by-class is realistic, testing robustness when test instances belong to classes not seen during training.\"}]","Selecting SAT Encodings - Constraints | PDF",1791081811,91]