[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84908-en":3,"doc-seo-84908-105":30,"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":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":13,"seo_description":14,"update_tm":28,"read_time":29},84908,1099514068035,"Ezra","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","Faster Exponential-Time Approximate Counting via Bounded Self-Reductions","Faster exponential-time randomized approximation algorithms target counting problems where polynomial-time approximation is unavailable and exact exponential-time counting is too costly. For n-vertex graphs, the independent-set counter runs in O*(1.1869^n) time, improving the prior O*(1.2041^n) general-graph bound. For n-variable #2-SAT, an O*(1.2373^n) approximation algorithm is derived, slightly below Wahlström’s cited O*(1.2377^n) variable-parameter exact bound.","Faster Exponential-Time Approximate Counting via Bounded  \nSelf-Reductions  \nKatie Clinch University of Queensland, Australia [k. clinch@uq. edu. au](k. clinch@uq. edu. au)  \nSimon Mackenzie UNSW Sydney, Australia [simon. william. mackenzie@gmail. com](simon. william. mackenzie@gmail. com)  \nSerge Gaspers UNSW Sydney, Australia  \n[serge. gaspers@unsw. edu. au](serge. gaspers@unsw. edu. au)[ ](serge. gaspers@unsw. edu. au)Qi Wang UNSW Sydney, Australia [wangqi118010296@outlook. com](wangqi118010296@outlook. com)  \narXiv :2607 .06393v 1 [ cs .DS] 7 Jul 2026  \nAbstract  \nWe give faster exponential-time randomised approximation algorithms for counting problems where polynomial-time approximation is unavailable and exact exponential-time counting remains expensive. For general n-vertex graphs, our independent-set counter runs in O∗ (1 .1869n ) time, improving the previous O ∗ (1 .2041n ) general-graph bound. For n-variable \\#2-SAT, we obtain an O ∗ (1 .2373n )-time approximation algorithm, narrowly below Wahlstr¨om’s currently cited O ∗ (1 .2377n ) variable-parameter exact bound.  \nThe new algorithmic point is to take the square root after decomposition. For a single bounded unweighted self-reduction with f (x) positive leaves and recursion-compatible upper bound b (x), an enumerate-or-sample estimator gives an (ε,δ)-approximation in  \nO ∗ 􀀐 pb (x) ε −2 log ~~1~~δ􀀑  \ntime. After preprocessing decomposes an input into many bounded cores, the combined estimator pays  \nO∗ sXi bi (xi) ε −2 log  ~~ 1~~δ ,   \nrather than estimating the cores separately at cost Pi p bi (xi) .  \nThe same conversion improves the bases for counting maximal cliques, minimal separators, and perfect matchings in subcubic graphs. Bounded unweighted self-reductions provide the formal language; at the level of counting classes, the resulting unweighted formulation has the same Karp closure as TotP. With explicit recursion-tree access, the framework yields black-box quantum speed-ups.  \n1 Introduction  \nCounting the solutions to combinatorial problems is a fundamental challenge in algorithms and complexity theory. The systematic study of counting problems from a computational complexity perspective may be said to have properly started in 1979 when Valiant introduced the complexity class \\#P [47] .  \nCounting problems arise across numerous domains, from determining the number of satisfying assignments of a logical formula (\\#SAT [11, 17], \\#2-SAT [14, 22 , 49], \\#DNF [30, 32]) to counting structures in graphs (independent sets [20, 25], matchings [17], maximal cliques [36], cycles [2],  \narbitrary subgraphs [48], colourings [19]), with applications such as evaluating partition functions in statistical physics [23, 28] . Because exact counting is intractable in general (\\#P-complete even for 2-CNF formulas [49], or computing the volume of a convex body [35]), research has also studied approximate counting. Fully Polynomial Randomised Approximation Schemes (FPRASs) output estimates within specified relative error with high probability in time polynomial in the input size and the inverse error parameters [30, 31]; FPTASs are deterministic analogues [27, 34] . Early successes include FPRASs for DNF counting [31], volume estimation for convex bodies [16], and the permanent of a non-negative matrix [29] .  \nFor self-reducible problems, a class that includes many natural counting problems, the existence of an approximation algorithm that achieves a polynomial factor error implies the existence of an FPRAS [42] . Furthermore, for such problems, almost uniform generation is inter-reducible with approximate counting, indicating that they are of similar complexity [30] . In spite of this, some counting problems remain intractable even in the approximation regime. For instance, a multiplicative approximator for \\#SAT can distinguish zero satisfying assignments from at least one satisfying assignment, and hence would decide SAT; therefore \\#SAT has no FPRAS unless NP = RP [17] .","cbCaimHUG93bOd2N","https://ap.wps.com/l/cbCaimHUG93bOd2N","pdf",668367,2,1,65,"English","en",105,"# Abstract\n# Introduction\n## Complexity of counting and approximation\n## Self-reducible counting problems\n## Main results and applications\n## Central estimator and bound-based sampling","[{\"question\":\"What counting problems does the paper focus on?\",\"answer\":\"The paper targets counting problems where polynomial-time approximation is unavailable and exact exponential-time counting is expensive, with main applications to #Independent-Set and #2-SAT.\"},{\"question\":\"What runtime improvements are achieved for independent sets and #2-SAT?\",\"answer\":\"For independent sets on n-vertex graphs, it achieves O*(1.1869^n), improving the prior general-graph O*(1.2041^n) bound. For n-variable #2-SAT, it achieves O*(1.2373^n), improving on the cited O*(1.2377^n) variable-parameter exact bound by Wahlström.\"},{\"question\":\"What is the key algorithmic idea behind the faster approximation?\",\"answer\":\"The approach decomposes the problem, then uses an enumerate-or-sample estimator that pays with the square-root after decomposition, combining estimators across bounded cores using recursion-compatible upper bounds.\"}]",1784199288,164,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"faster-exponential-time-approximate-counting-via-bounded-self-reductions","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,47,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":20},"https://docshare.wps.com/document/","Document",{"item":48,"name":12,"@type":43,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/faster-exponential-time-approximate-counting-via-bounded-self-reductions/84908/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-22","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 counting problems does the paper focus on?","Question",{"text":75,"@type":76},"The paper targets counting problems where polynomial-time approximation is unavailable and exact exponential-time counting is expensive, with main applications to #Independent-Set and #2-SAT.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What runtime improvements are achieved for independent sets and #2-SAT?",{"text":80,"@type":76},"For independent sets on n-vertex graphs, it achieves O*(1.1869^n), improving the prior general-graph O*(1.2041^n) bound. For n-variable #2-SAT, it achieves O*(1.2373^n), improving on the cited O*(1.2377^n) variable-parameter exact bound by Wahlström.",{"name":82,"@type":73,"acceptedAnswer":83},"What is the key algorithmic idea behind the faster approximation?",{"text":84,"@type":76},"The approach decomposes the problem, then uses an enumerate-or-sample estimator that pays with the square-root after decomposition, combining estimators across bounded cores using recursion-compatible upper bounds.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]