[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85281-en":3,"doc-seo-85281-105":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},85281,1374391974585,"Genevieve","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","Threshold Rounding and Bounded-Degree Boolean MAX 2-CSP","The paper studies how to improve approximation guarantees for Maximum Constraint Satisfaction Problems (MAX CSP) when constraint-graph degree is bounded. Given a MAX CSP P with optimal approximation factor αP, it asks whether one can efficiently achieve αP + ϵd for instances with degree at most constant d, where ϵd depends on d. Building on a THRESH−-rounding framework over a canonical SDP, the work targets bounded-degree Boolean MAX 2-CSPs under a natural “reasonable” condition and establishes stronger approximation bounds.","arXiv :2607 . 1 1050v 1 [ cs .DS] 13 Jul 2026  \nThreshold Rounding and Bounded-Degree Boolean MAX 2-CSP  \nSuprovat Ghoshal∗ Neng Huang† Euiwoong Lee‡ Konstantin Makarychev§  \nYury Makarychev¶  \nJuly 14, 2026  \nAbstract  \nBothapopWoeptirmlecoeaasxadneimlesMofatapcAprMioribXnoe2AxaXaim-ln ΩCS2goatP-Sitio(1iAhn/dnsTmra,4taft)woi-ineromcebfprsimoorovinprundMemwhovedAXeie-ntchthde2-egoSeveverareeATrrtthyioMarevafuAXchshriartievobh2ele-lddeSr roapanATbyunpdtdeaoinsheingrsbttaLinanLscinceZhasetaa, memnwlgosos(βherio[bL(roanst1/dis tLZrda2h0i)e2cn)]lats-faUssctGOoInorCur-frewapsuithplrt gdeoxiengrmeearebasiliblitzeouys andrenesudl(αbts,(1tX/do HD2I-)sC)i-faehUctaTorndanapKdpotMrohaAxrXiim2[H-aAtKio2Nn3D]a[lgTBorogHPitehtZmhe23r]f,orwoMithurAXthreseuCstltUasTteug-oognfe-sgrthtsaetp-ahhrastt similar improvements exist for bounded-degree instances of these problems as well.  \n1 Introduction  \nMaximum Constraint Satisfaction Problems (MAX CSPs) are a fundamental class of combinatorial problems that are studied extensively in theoretical computer science. The design of approximation algorithms for MAX CSPs has been a central topic of research that has seen substantial progress in the past decades. Some of the main highlights include the design of approximation algorithms for MAX CSPs [MM17] and the development of a framework for hardness reductions to study the limits of approximation for CSPs [Kho05 , Kho10] . These developments have culminated in surprisingly tight characterizations of the optimal approximation factor of every MAX CSP [Rag08], under the Unique Games Conjecture (UGC) [Kho02] .  \nIn this work, we aim to explore the approximability of MAX CSPs for structurally simpler instances, namely, when the instances are sparse. Specifically, we explore the question of whether one can improve on the optimal approximation guarantee of a MAX CSP when the degree of the underlying constraint graph is bounded. Formally, we are interested in the following:  \nGiven a MAX CSP problem P, suppose its optimal approximation factor is αP . Then, if the degree of the instance is bounded by some constant d, can we efficiently achieve αP + ϵd approximation,  \nwhere the improvement ϵd depends on the degree d?  \nThe above question has been studied for various specific MAX CSPs in the literature [BK02 , HLZ04 ,  \n∗ Indiana University.  \n†University of Michigan.  \n‡University of Michigan. Supported in part by NSF award CCF-2236669 .  \n§ Northwestern University. Supported by NSF awards CCF-1955351 and EECS-2216970 .¶ Toyota Technological Institute at Chicago. Supported by NSF award EECS-2216899 .  \nFKL02 , AKS09 , BMO+15 , HK23] . Arguably, one of the most interesting lines of work among these involves the approximability of MAX CUT in this setting. It was first studied by Feige, Karpinski, and Langberg [FKL02], who showed that the MAX CUT problem on graphs with maximum degree d admits an efficient approximation guarantee of αGW + Ω(1/d4 ) , where αGW ≈ 0.87856 denotes the optimal approximation factor for MAX CUT under the UGC [GW94 , KKMO07] . This was then improved by Florén [Flo16] up to an αGW + Ω(1/d3 ) factor, and by Hsieh and Kothari [HK23] to afathboctatuonrodonefed-αcdrs/d2thancee)s. aApnpotrohxier notable wmation resisrkanicne thofisMcoAnXtexk-t iLIsNthat ofand oth[BMO+er rela5],ed wphroobsholemwseidn Unfortunately, generalizing this phenomenon beyond the specific examples mentioned above has been challenging. In particular, such results are not known even for well-studied friends of the MAX  \nCUT problem, such as MAX 2-SAT and MAX 2-AND. One of the main difficulties in obtaining similar guarantees for general MAX CSPs has been the following: most of the aforementioned improvements in the bounded-degree setting implicitly rely on tight structural characterizations of the hardest-to-round, i.e. , integrality gap, instances of the problem 1. Such explicit characterizations are known only for a handful of examples (e.g., MAX CUT [FS02], MAX k-LIN [H","cbCairhX8yXd22Yx","https://ap.wps.com/l/cbCairhX8yXd22Yx","pdf",595269,1,25,"English","en",105,"# Introduction\n## Our Results","[{\"question\":\"What problem does the paper address for MAX CSPs in bounded-degree graphs?\",\"answer\":\"It asks whether, for sparse (bounded-degree) instances, one can improve beyond the general optimal approximation factor αP by achieving αP + ϵd efficiently, with ϵd depending on the degree bound d.\"},{\"question\":\"What algorithmic framework does the paper build on?\",\"answer\":\"It studies a family of algorithms that solve a canonical SDP relaxation for the MAX CSP and then rounds the SDP solution using THRESH− rounding functions.\"},{\"question\":\"For which class of instances does the paper provide improved guarantees?\",\"answer\":\"The paper focuses on bounded-degree Boolean MAX 2-CSP instances that satisfy a natural “reasonable” condition, and derives improved approximation results for that setting.\"}]",1784202236,63,{"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},"threshold-rounding-and-bounded-degree-boolean-max-2-csp","",{"@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/threshold-rounding-and-bounded-degree-boolean-max-2-csp/85281/",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 address for MAX CSPs in bounded-degree graphs?","Question",{"text":75,"@type":76},"It asks whether, for sparse (bounded-degree) instances, one can improve beyond the general optimal approximation factor αP by achieving αP + ϵd efficiently, with ϵd depending on the degree bound d.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What algorithmic framework does the paper build on?",{"text":80,"@type":76},"It studies a family of algorithms that solve a canonical SDP relaxation for the MAX CSP and then rounds the SDP solution using THRESH− rounding functions.",{"name":82,"@type":73,"acceptedAnswer":83},"For which class of instances does the paper provide improved guarantees?",{"text":84,"@type":76},"The paper focuses on bounded-degree Boolean MAX 2-CSP instances that satisfy a natural “reasonable” condition, and derives improved approximation results for that 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