[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81628-en":3,"doc-seo-81628-105":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},81628,8796095360427,"Lucas Martin","https://ap-avatar.wpscdn.com/davatar_994ba38a5ba835b3df7d355c54d3ed8d",8,"Research & Report","Improved space-time tradeoff for TSP via extremal set systems","The traveling salesman problem (TSP) is a central challenge in combinatorial optimization, with longstanding exact algorithms whose space–time bounds have resisted improvement. This work establishes a strictly better space–time tradeoff for all 2 \u003C T \u003C 4, achieving a minimum of ST \u003C 3.572. The approach builds sparse set systems (hypergraphs) admitting many maximal chains, while also disproving a 2013 conjecture connecting previous tradeoff optimality for TSP.","arXiv :2604 .05645v 1 [ cs .DS] 7 Apr 2026  \nImproved space-time tradeoff for TSP via extremal set systems  \nJustin Dallant 1 and László Kozma 1  \n1 Faculty of Computer Science, TU Dresden, Germany  \nAbstract  \nThe traveling salesman problem (TSP) is a cornerstone of combinatorial optimization and has deeply influenced the development of algorithmic techniques in both exact and approximate settings. Yet, improving on the decades-old bounds for solving TSP exactly remains elusive:  \nthe dynamic program of Bellman, Held, and Karp from 1962 uses 2n+O(log n) time and space, and the divide-and-conquer approach of Gurevich and Shelah from 1987 uses 4n+O(log2 n) time and polynomial space. A straightforward combination of the two algorithms trades off Tn+o(n) time and Sn+o(n) space at various points of the curve ST = 4 . An improvement to this tradeoff when 2 \u003C T \u003C 2 √2 was found by Koivisto and Parviainen (SODA 2010), yielding a minimum of ST ≈ 3.93. Koivisto and Parviainen show their method to be optimal among a broad class of partial-order-based approaches, and to date, no improvement or alternative method has been found.  \nIn this paper we give a tradeoff that strictly improves all previous ones for all 2 \u003C T \u003C 4 , achieving a minimum of ST \u003C 3.572. A key ingredient is the construction of sparse set systems (hypergraphs) that admit a large number of maximal chains. The existence of such objects is of independent interest in extremal combinatorics, likely to see further applications. Along the way we disprove a combinatorial conjecture of Johnson, Leader, and Russell from 2013, relating it with the optimality of the previous tradeoff schemes for TSP. Our techniques extend to a broad class of permutation problems over arbitrary semirings, yielding improved space–time tradeoffsin these settings as well.  \n1. Introduction  \nThe traveling salesman problem (TSP) asks, given a set of n cities S = {c1 , c2 ,..., cn} and distances d : S2 → R, for a tour of minimal length that visits each city in S exactly once. More precisely, we seek a permutation π of [n] = {1, 2 ,..., n} that minimizes  \nn−1  \nd (c π(n), c π(1)) +X d(c π(i), c π(i+1)) .  \ni=1  \nTSP is an emblematic problem of computer science that has been investigated since the pioneering era of the field [Coo11] . The fascination it holds is likely due to multiple factors such as its clear and intuitive statement, its practical relevance and important special cases (e.g. , metric, Euclidean), and the fact that it can be attacked with a variety of algorithmic techniques. The question of the  \n1 Email: justin.dallant@tu-dresden.de, laszlo.kozma@tu-dresden.de  \nalgorithmic complexity of TSP can be traced back to Menger in the 1920s [Sch05], and the problem continues to inspire research to date, in both exact and approximate settings.  \nThe 1962 dynamic programming algorithm of Bellman [Bel62] and Held and Karp [HK62] solves TSP by observing that any set of cities that appear contiguously at the beginning of the optimal tour must themselves be visited in an optimal order (as otherwise the entire tour could be improved) . The algorithm then computes, for all subsets S′ ⊆ S of cities, i.e. , all possible prefix-sets of the optimal tour, the shortest way of visiting S′ from a given start-to a given endpoint. Both the timeand the space requirement 1 is O∗ (2n ), dominated by the number of subsets of S ; we recall this algorithm in more detail in § 2.  \nImproving the running time to O∗(cn ) with c \u003C 2 has been a notorious open problem for well over half a century. For another prototypical NP-hard problem, CNF-SAT, the impossibility of such an improvement is the subject of the Strong Exponential-Time Hypothesis (SETH) [IP01] . Breaking the 2n-barrier for TSP would similarly be a major breakthrough. Such improvements have been achieved only in special cases, most notably for the unweighted, undirected case, i.e., for Hamiltonicity: here, an algorithm with running time O∗ (1 .66n ) was found by B","cbCaifIIFR0lyfl5","https://ap.wps.com/l/cbCaifIIFR0lyfl5","pdf",790222,4,1,22,"English","en",105,"# Abstract\n# Introduction","[{\"question\":\"What improvement does the paper provide for the exact Traveling Salesman Problem (TSP)?\",\"answer\":\"It introduces a space–time tradeoff that strictly improves all previous ones for every 2 \\u003c T \\u003c 4, with a minimum value ST \\u003c 3.572.\"},{\"question\":\"What is the key technical ingredient used to achieve the new tradeoff?\",\"answer\":\"The method constructs sparse set systems (hypergraphs) that admit a large number of maximal chains, which is highlighted as independently interesting in extremal combinatorics.\"},{\"question\":\"Which earlier claim does the paper challenge or refute?\",\"answer\":\"It disproves a combinatorial conjecture of Johnson, Leader, and Russell from 2013 that relates to the optimality of earlier tradeoff schemes for 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improvement does the paper provide for the exact Traveling Salesman Problem (TSP)?","Question",{"text":75,"@type":76},"It introduces a space–time tradeoff that strictly improves all previous ones for every 2 \u003C T \u003C 4, with a minimum value ST \u003C 3.572.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What is the key technical ingredient used to achieve the new tradeoff?",{"text":80,"@type":76},"The method constructs sparse set systems (hypergraphs) that admit a large number of maximal chains, which is highlighted as independently interesting in extremal combinatorics.",{"name":82,"@type":73,"acceptedAnswer":83},"Which earlier claim does the paper challenge or refute?",{"text":84,"@type":76},"It disproves a combinatorial conjecture of Johnson, Leader, and Russell from 2013 that relates to the optimality of earlier tradeoff schemes for 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