[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-156872-105":59,"doc-detail-156872-en":131},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":124,"head_meta":126,"extra_data":128,"updated_unix":130},105,"en","uniform-solution-sampling-using-a-constraint-solver-as-an-oracle","Uniform Solution Sampling Using a Constraint Solver As an Oracle","","Hard constraints define feasible solutions in combinatorial spaces, making uniform sampling a difficult problem closely tied to model counting. The work introduces a black-box sampling technique that preserves uniform exploration while leveraging the reasoning power of systematic constraint solvers. Through challenging domains such as energy barriers and highly asymmetric landscapes, the study shows standard methods like Simulated Annealing and Gibbs Sampling degrade substantially. The proposed scheme also yields a new approximate model counting approach with strong empirical accuracy on benchmark problems.",{"@graph":69,"@context":123},[70,84,106],{"@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/uniform-solution-sampling-using-a-constraint-solver-as-an-oracle/156872/",{"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/uniform-solution-sampling-using-a-constraint-solver-as-an-oracle/156872.png","ImageObject",300,407,{"name":92,"@type":93},"Rhys","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-06","2026-08-29",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",13,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"What problem does the paper focus on?","Question",{"text":113,"@type":114},"It focuses on sampling uniformly from solutions defined by hard constraints in combinatorial spaces, where reaching a uniform distribution is difficult and often computationally expensive.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"Why do standard sampling methods struggle with hard constraints?",{"text":118,"@type":114},"Hard constraints introduce rugged or highly asymmetric energy landscapes, which can cause long mixing times or biased stationary distributions for methods such as Simulated Annealing and Gibbs Sampling.",{"name":120,"@type":111,"acceptedAnswer":121},"How does SearchTreeSampler address the uniform sampling challenge?",{"text":122,"@type":114},"SearchTreeSampler uses a systematic constraint solver as a black box to guide exploration while enforcing (approximately) uniform coverage of the search space, reducing reliance on heuristically chosen proposal distributions.","https://schema.org",{"og:url":83,"og:type":125,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":127,"canonical":83},"index,follow",{"doc_id":129,"site_id":62},156872,1787968197,{"code":4,"msg":5,"data":132},{"doc_id":129,"user_id":133,"nickname":92,"user_avatar":134,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":52,"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":130,"read_time":144,"preview_page_count":4},687207024643,"https://ap-avatar.wpscdn.com/davatar_3d24733baf745e90a7e4bdd5f77d97b2","Uniform Solution Sampling Using a Constraint Solver As an Oracle∗  \nStefano Ermon  \nDepartment of Computer Science Cornell University [ermonste@cs.cornell.edu](ermonste@cs.cornell.edu)  \nCarla Gomes  \nDepartment of Computer Science Cornell University [gomes@cs.cornell.edu](gomes@cs.cornell.edu)  \nBart Selman  \nDepartment of Computer Science Cornell University [selman@cs.cornell.edu](selman@cs.cornell.edu)  \nAbstract  \nWe consider the problem of sampling from solutions defined by a set of hard constraints on a combinatorial space. We propose a new sampling technique that, while enforcing a uniform exploration of the search space, leverages the reasoning power of a systematic constraint solverin a black-box scheme. We present a series of challenging domains, such as energy barriers and highly asymmetric spaces, that reveal the difficulties introduced by hard constraints.  \nWe demonstrate that standard approaches such as Simulated Annealing and Gibbs Sampling are greatly affected, while our new technique can overcome many of these difficulties. Finally, we show that our sampling scheme naturally definesa new approximate model counting technique, which we empirically show to be very accurate on a range of benchmark problems.  \n1 Introduction  \nIn recent years, we have seen significant interest in probabilistic reasoning approaches that combine both hard, logical constraints and probabilistic information through soft, weighted constraints. Markov Logic Networks [1] are a prominent example of such a modeling language. The hard constraints are used to capture definitional and other irrefutable relationships in the underlying domain, while soft constraints capture less categorical information, generally better for modeling real-world data and dependencies.  \nMarkov Chain Monte Carlo (MCMC) methods, such as Simulated Annealing (SA) and Gibbs Sampling [2], are among the most prominent approaches to probabilistic reasoning, especially when exact inference is beyond reach. In fact, they are guaranteed to asymptotically converge to  \n∗ This work was supported by NSF Grant 0832782 .  \na stationary distribution that can theoretically provide uniform samples. However, sampling from the solutions of a set of hard constraints is believed to be very hard in worst case, as it is closely related to \\#-P complete problems such as model counting [3, 4, 5, 6, 7] . In particular, the time required for a Markov Chain to reach the stationary distribution (mixing time) is often exponential in the number of problem variables when hard constraints are present. These difficulties led to the introduction of alternate sampling strategies. For example, in the SampleSAT approach [8], a Markov Chain is constructed that combines moves proposed according to a Simulated Annealing Markov Chain with so-called WalkSAT moves, inspired by local search constraint solvers. The SampleSAT method provides a significant advance, since SA sampling on constraint problems of practical interest generally does not reach any solutions or, at best, only a small subset of all possible solutions. SampleSAT was subsequently incorporated into MC-SAT in the Alchemy package for reasoning and learning for Markov Logic Networks [9] . A key limitation of SampleSAT is that it is not guaranteed to converge to the uniform distribution on the solution set. Infact, one can create examples where the stationary distribution of SampleSAT is arbitrarily biased (non-uniform) in terms of solution samples. Still, this does not negate the value of SampleSAT in many practical settings. For, although SA-like sampling does sample in the limit from the uniform stationary distribution on the solution set, reaching the stationary distribution often requires exponential time, making the SA strategy of little use in practice.  \nIn this paper, we revisit the question of how to devise practical methods for sampling from solutions defined by a set of hard constraints on a combinatorial space. We present a series of challeng","cbCaivclyUji443N","https://ap.wps.com/l/cbCaivclyUji443N","pdf",346462,"English","# Introduction\n## Probabilistic reasoning with hard and soft constraints\n## MCMC approaches and limitations with hard constraints\n## Energy landscapes and structured opportunities\n## SearchTreeSampler overview","[{\"question\":\"What problem does the paper focus on?\",\"answer\":\"It focuses on sampling uniformly from solutions defined by hard constraints in combinatorial spaces, where reaching a uniform distribution is difficult and often computationally expensive.\"},{\"question\":\"Why do standard sampling methods struggle with hard constraints?\",\"answer\":\"Hard constraints introduce rugged or highly asymmetric energy landscapes, which can cause long mixing times or biased stationary distributions for methods such as Simulated Annealing and Gibbs Sampling.\"},{\"question\":\"How does SearchTreeSampler address the uniform sampling challenge?\",\"answer\":\"SearchTreeSampler uses a systematic constraint solver as a black box to guide exploration while enforcing (approximately) uniform coverage of the search space, reducing reliance on heuristically chosen proposal distributions.\"}]","Uniform Solution Sampling Using a Constraint Solver As an Oracle | PDF",25]