[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82339-en":3,"doc-seo-82339-105":29,"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":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},82339,1374391974585,"Genevieve","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","Strong Refutation of Random Ordering CSPs","This work initiates the study of strong refutation for random ordering constraint satisfaction problems (CSPs). It presents a polynomial-time ε-refutation algorithm for random ordering CSPs with predicate P when clause counts exceed a threshold depending on the coordinate degree d and ε. The paper derives a smooth three-way tradeoff among runtime, clause density, and refutation strength ε via the Kikuchi method. It further establishes a computational lower bound for low coordinate degree algorithms, showing the tradeoff is near optimal.","arXiv :2607 .094 10v 1 [ cs .DS] 10 Jul 2026  \nStrong Refutation of Random Ordering CSPs  \nXifan Yu∗  \nDepartment of Computer Science, Yale University  \nJuly 13, 2026  \nAbstract  \nIn this work, we initiate the study of strongly refuting the satisfiability of random ordering constraint satisfaction problems. We show that there is a polynomial-time ε-refutation algorithm for random ordering CSP with predicate P when the number of clauses is above the threshold ˜Ω 􀀐nd/2/ε2 􀀑 , where d is the coordinate degree of the predicate P. We further give a smooth three-way tradeoff between the running time, the clause density, and the refutation strength ε using the Kikuchi method. Finally, we complement our algorithmic results with a computational lower bound based on the class of low coordinate degree algorithms, providing evidence that the established three-way tradeoff is near optimal.  \n1 Introduction  \nConstraint satisfaction problems (CSPs) have been a central object in theoretical computer science, with notable examples such as 3-SAT, MAX-CUT, and graph coloring. There is a long line of research on CSPs, ranging from worst-case approximability [Rag08], algebraic properties of CSP predicates [BJK05, Bul17, Zhu20], and fine-grained complexity [IP01] .  \nBeyond worst-case analysis, there is also a vast literature studying the average-case complexity of CSPs, which has received a lot of attention due to its connection to hardness of approximation [Fei02], proof complexity [BSB02], learning theory [DLSS14], cryptography [ABW10], and statistical physics [AP03] .  \nCSP Refutation. A natural average-case problem associated to random CSPs is the refutation task. Statistically, a random k-CSP is known to be unsatisfiable with high probability once its clause density ~~m~~n exceeds a large constant depending on k. For k-SAT, it is moreover conjectured that there exists a constant αk such that a random k-SAT is unsatisfiable with high probability above the clause density threshold ~~m~~n > αk , and satisfiable with high probability below the clause density threshold ~~m~~n \u003C αk . This conjecture has been settled by [DSS15] for all large enough k.  \nHowever, given a random CSP instance with large enough clause density, can we efficiently refute its satisfiability, i.e., to certify that the instance is unsatisfiable or to declare failure when we cannot produce such a certificate efficiently? Importantly, a refutation algorithm cannot “cheat” by always outputting UNSAT, because there is a tiny chance that a random CSP instance with high clause density is still satisfiable and the algorithm should not make any mistakes on such instances. The goal of a refutation algorithm is therefore to efficiently provide a certificate of unsatisfiability with high probability over the distribution of random CSP. A closely related problem is called the strong refutation task, where the goal is not only to with high probability refute the satisfiability of a random CSP, but also to certify that no assignments can satisfy many more clauses than what a random assignment achieves in expectation.  \nWe remark that the strong refutation task is in stark contrast to the search/optimization task, since a strong refutation algorithm needs to certify that any assignment cannot satisfy too many clauses whereas the search task only needs to find one good assignment.  \n∗ Email: [xifan.yu@yale.edu](xifan.yu@yale.edu. Partially)[. Partially](xifan.yu@yale.edu. Partially) supported by ONR Award N00014-24-1-2611.  \nFor k-XOR, there is a natural spectral algorithm which can strongly refute a random k-XOR instance efficiently above the density m ≥ Ω(nk/2 poly log(n)) [AOW15] . When the clause density is below the spectral threshold n ≪ m ≪ nk/2, even though no polynomial-time strong refutation algorithm is known (and it is conjectured that efficient strong refutation is impossible below the threshold nk/2), there are subexponentialtime algorithms [RRS17], which run in time exp(nδ )","cbCaiqBjl73DPTpY","https://ap.wps.com/l/cbCaiqBjl73DPTpY","pdf",605521,1,52,"English","en",105,"# Abstract\n# Introduction\n## CSP Refutation\n## Ordering CSPs","[{\"question\":\"What problem does the paper study in random ordering CSPs?\",\"answer\":\"The paper studies strong refutation for random ordering constraint satisfaction problems, aiming to certify unsatisfiability with high probability and to limit how many clauses any assignment can satisfy beyond random expectation.\"},{\"question\":\"What main algorithmic result is proved?\",\"answer\":\"It gives a polynomial-time ε-refutation algorithm for random ordering CSPs under a clause-density condition above a threshold that depends on the predicate’s coordinate degree d and on ε.\"},{\"question\":\"How is the tradeoff between runtime, density, and refutation strength obtained and validated?\",\"answer\":\"A smooth three-way tradeoff between running time, clause density, and ε is derived using the Kikuchi method, and the paper supports near-optimality with a computational lower bound for low coordinate degree algorithms.\"}]",1784179754,131,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"strong-refutation-of-random-ordering-csps","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/strong-refutation-of-random-ordering-csps/82339/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","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 problem does the paper study in random ordering CSPs?","Question",{"text":75,"@type":76},"The paper studies strong refutation for random ordering constraint satisfaction problems, aiming to certify unsatisfiability with high probability and to limit how many clauses any assignment can satisfy beyond random expectation.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What main algorithmic result is proved?",{"text":80,"@type":76},"It gives a polynomial-time ε-refutation algorithm for random ordering CSPs under a clause-density condition above a threshold that depends on the predicate’s coordinate degree d and on ε.",{"name":82,"@type":73,"acceptedAnswer":83},"How is the tradeoff between runtime, density, and refutation strength obtained and validated?",{"text":84,"@type":76},"A smooth three-way tradeoff between running time, clause density, and ε is derived using the Kikuchi method, and the paper supports near-optimality with a computational lower bound for low coordinate degree 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