[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81630-en":3,"doc-seo-81630-105":30,"detail-sidebar-cat-0-en-105":83},{"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},81630,8796095360427,"Lucas Martin","https://ap-avatar.wpscdn.com/davatar_994ba38a5ba835b3df7d355c54d3ed8d",8,"Research & Report","Improved Space-Time Tradeoffs for Permutation Problems via Extremal Combinatorics","Improved space-time tradeoffs are presented for permutation problems over additively idempotent semi-rings, including an algorithm for the Traveling Salesperson Problem on N vertices with space S and time T such that S·T ≤ 3.1861N. The work strengthens Koivisto and Parviainen by replacing the prior guarantee of 3.9271N and surpasses their stated barrier. A new set-system parameter, chain efficiency, links maximal chains and system cardinality, and it yields efficient tradeoffs and new constructions that disprove a conjecture.","arXiv :2604 .0566 1v2 [ cs .DS] 10 Jul 2026  \nImproved Space-Time Tradeoffs for Permutation Problems via  \nExtremal Combinatorics∗  \nAfrouz Jabal Ameli† Jesper Nederlof† Shengzhe Wang†  \nAbstract  \nWe provide improved space-time tradeoffs for permutation problems over additivelyidempotent semi-rings. In particular, there is an algorithm for the Traveling Salesperson Problem that solves N-vertex instances using space S and time T where S · T ≤ 3.1861N . This improves a previous work by Koivisto and Parviainen [SODA’10] where S · T ≤ 3.9271N , and overcomes a barrier they identified, as their bound was shown to be optimal within their framework.  \nTo get our results, we introduce a new parameter of a set system that we call the chain efficiency. This relates the number of maximal chains contained in the set system with the cardinality of  \nthe system. We show that set systems of high efficiency imply efficient space-time tradeoffs for permutation problems, and give constructions of set systems with high chain efficiency, disproving a conjecture by Johnson, Leader and Russell [Comb. Probab. Comput.’15] .  \n1 Introduction  \nThe area of exact exponential time algorithms (or fine-grained complexity of NP-complete problems) asks for the most fundamental NP-complete problems how efficiently they can be solved exactly in the worst case. While early papers initiated this endeavor already in the 20th century [4, 17, 18], the area flourished around 2010 thanks to influential surveys by Woeginger [31, 32] . There is also a textbook on the topic [13], and several more recent surveys [12, 27] .  \nBesides the main goal of fast running times there has also consistently been major emphasis on designing algorithms with minimum space usage. Space usage is already prevalent in (even the title of) the initial survey [31], and the space usage of the aforementioned classical results was improved already many years ago [29, 24], while further progress continues to be made for certain problems [3, 5, 28] .  \nOne of the most central computational problems studied in the area of exact exponential time algorithms is the Traveling Salesperson Problem (TSP), and it was already a topic of very early work [4, 17] that solves N-vertex instances in O ∗ (2N ) time.1 It poses the following very ambitious question:  \nOpen Problem 1 (Problem 6 in [31]).  \n(a) Find an algorithm that solves N-vertex TSP in O ∗ (1 .99N ) time,  \n(b) Find an algorithm that solves N-vertex TSP in O ∗ (2N ) time and polynomial space.  \n∗ All authors are supported by the project COALESCE that has received funding from the European Research Council (ERC), grant agreement No 853234 .  \n†Department of Information and Computing Sciences, Utrecht University, The Netherlands. [Email addresses:](Email addresses: a. jabalameli@uu. nl)[ a. jabalameli@uu. nl](Email addresses: a. jabalameli@uu. nl), [j. nederlof@uu. nl](j. nederlof@uu. nl), [s. wang5@uu. nl](s. wang5@uu. nl)  \n[1](1The O)[The](1The O)[ O](1The O)∗ (·) notation omits factors polynomial in the input size.  \nWhile some progress on Open Problem 1(a) in the unweighted and bipartite cases has been made [6, 26], it still remains wide open. For Open Problem 1(b), [15] gave an O ∗ (4N ) time and poly space algorithm2 , and it is a well-known open question whether this can be improved to even O ∗ (3 .99N ) time and polynomial space.  \nSpace-Time Tradeoffs. Instead of requiring polynomial space as done in Open Problem 1(b), one can also study time-space trade-offs: For every S = S (N), what is the fastest running time T = T (N) of any algorithm that solves N-vertex TSP using only S space? Such space-time tradeoffs are studied for several other NP-complete problems, like for example the Subset Sum problem [2, 10] .  \nThe algorithms of [4, 17] and [15] can be mixed to get, for every i = 0 , . . . , ⌈log2 N⌉ an algorithm that runs in S = O ∗ (2N/2i ) space and T = O ∗ (2 (2−1/2i )N ) time (confer [13]) and one may expect that improving over the t","cbCaiiv1mXgV3Uwq","https://ap.wps.com/l/cbCaiiv1mXgV3Uwq","pdf",684034,3,1,25,"English","en",105,"# Abstract\n# Introduction\n## Exact exponential time algorithms and space\n## Traveling Salesperson Problem and open problems\n## Space-time tradeoffs and permutation problems\n# Problem setup over semi-rings\n## Definition of permutation problems (bounded degree)\n# Main theorem and proof strategy","[{\"question\":\"Which conjecture is disproved and how?\",\"answer\":\"The paper constructs set systems with high chain efficiency that disprove a conjecture by Johnson, Leader and Russell (Comb. Probab. Comput.’15).\"}]",1784174972,63,{"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":78,"head_meta":80,"extra_data":82,"updated_unix":28},"improved-space-time-tradeoffs-for-permutation-problems-via-extremal-combinatorics","",{"@graph":36,"@context":77},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":20},"https://docshare.wps.com/document/research-report/",{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/improved-space-time-tradeoffs-for-permutation-problems-via-extremal-combinatorics/81630/",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","2026-07-25","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71],{"name":72,"@type":73,"acceptedAnswer":74},"Which conjecture is disproved and how?","Question",{"text":75,"@type":76},"The paper constructs set systems with high chain efficiency that disprove a conjecture by Johnson, Leader and Russell (Comb. 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