[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82289-en":3,"doc-seo-82289-105":29,"detail-sidebar-cat-0-en-105":95},{"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},82289,13056703019662,"Evangeline","https://ap-avatar.wpscdn.com/avatar/be000253a8e92610077?_k=1778726343310543188",8,"Research & Report","Faster Exact Algorithms for Equal-Subset-Sum","Exact algorithms for Equal-Subset-Sum (ESS) are analyzed in the worst case: given a set S of n integers, two distinct subsets A and B with equal total sums are sought. A new state of the art improves the fastest known Randolph and Węgrzycki algorithm from O*(1.7067^n) time/space to O*(1.6994^n) time using O*(1.5664^n) space. Polynomial-space performance is also improved from O*(2.6817^n) to O*(2.5430^n). Time-space tradeoffs are studied and strengthened.","Faster Exact Algorithms for Equal-Subset-Sum  \nRyosuke Yamano∗ Tetsuo Shibuya†  \narXiv :2607 .09289v 1 [ cs .DS] 10 Jul 2026  \nAbstract  \nWe study exact algorithms for Equal-Subset-Sum in the worst-case setting: given a set S of n integers, find two distinct subsets A, B ⊆ S whose sums are equal. We establish a new stateof-the-art bound for this problem by improving the fastest known algorithm, due to Randolph and Węgrzycki (STOC 2026), from O ∗ (1 .7067n ) time and space to an algorithm that runs in O ∗ (1 .6994n ) time and uses O ∗ (1 .5664n ) space. We also improve the best known polynomialspace running time, due to Mucha, Nederlof, Pawlewicz, and Węgrzycki (ESA 2019), from O ∗ (2 .6817n ) to O ∗ (2 .5430n ) . Finally, we investigate time-space tradeoffs for this problem and improve the running times achievable under a broad range of exponential-space bounds.  \n∗ Department of Computer Science, Graduate School of Information Science and Technology, The University of Tokyo, [Japan. ryoyamano15@g.ecc.u-tokyo.ac.jp](Japan. ryoyamano15@g.ecc.u-tokyo.ac.jp).  \n†Division of Medical Data Informatics, Human Genome Center, Institute of Medical Science, The University of Tokyo, [Japan. tshibuya@hgc.jp. Supported](Japan. tshibuya@hgc.jp. Supported) by KAKENHI Grant Number 23K28035.  \n1 Introduction  \nSubset-Sum is a fundamental NP-hard problem, which can be formulated as follows.  \nProblem 1.1 (Subset-Sum) . Given a multiset S of n integers and a target integer t, output a subset A ⊆ S such that t = Pa∈A a, if such a subset exists.  \nSubset-Sum admits a simple meet-in-the-middle algorithm, due to Horowitz and Sahni [19], that runs in O∗ (2n/2) time 1. This remains the standard running-time bound for exact algorithms for Subset-Sum, and the fastest known algorithm [13] improves over the O(2n/2) bound only by a polynomial factor. Whether there exists an algorithm for Subset-Sum that runs in O∗ (2(0 .5−δ)n )  \ntime for some constant δ > 0 remains a long-standing open question.  \nClosely related to this open problem, Howgrave-Graham and Joux [20] broke this meet-in-themiddle barrier in the average-case setting by giving an O∗ (20 .337n )-time algorithm, which was later improved to O∗ (20 .283n ) by [8, 9] . The algorithm of Howgrave-Graham and Joux introduced the representation technique, which was later also used for Equal-Subset-Sum (ESS), an important variant of Subset-Sum that can be formulated as follows.  \nProblem 1.2 (Equal-Subset-Sum (ESS) [31]) . Given a set S of n integers, output two distinct subsets A, B ⊆ S such that Pa∈A a = Pb∈B b, if such subsets exist.  \nESS admits a simple meet-in-the-middle algorithm that runs in O∗ (3n/2) time. Surprisingly, the authors of [25] broke this meet-in-the-middle barrier for ESS in the worst-case setting by exploiting the representation technique, giving an algorithm that runs in O∗ (1 .7088n ) time and space. In the average-case setting, the authors of [12] gave an O∗ (30 .387n ) ≤ O∗ (1 .5299n )-time algorithm for ESS. The open question of whether the running time for ESS in the worst-case setting can be improved beyond O∗ (1 .7088n ) [21, Section 5, Question 1] was answered affirmatively very recently by Randolph and Węgrzycki [28], who gave an algorithm that runs in O∗ (1 .7067n ) time and space.  \n1.1 Our Results  \nIn this work, we focus on exact algorithms for ESS in the worst-case setting. All our algorithms are Monte Carlo algorithms that never return false positives, and their error probabilities can be reduced to 2−Ω(n) by repetition. Our first result establishes a new state-of-the-art algorithm for ESS by further improving the result of [28] . While the running-time improvement is our primary contribution, the algorithm also reduces the space usage by a much larger exponential factor.  \nTheorem 1.3 . There exists a Monte Carlo algorithm for ESS that runs in O∗ (1 .6994n ) time and uses O∗ (1 .5664n ) space.  \nOur second result improves the polynomial-space running time for ESS. We","cbCaibrZIsBlqTUN","https://ap.wps.com/l/cbCaibrZIsBlqTUN","pdf",679288,1,27,"English","en",105,"# Introduction\n## Our Results\n# Time-Space Tradeoffs","[{\"question\":\"What problem does the document address?\",\"answer\":\"The document studies Equal-Subset-Sum (ESS), where two distinct subsets of a given integer set must have equal sums in the worst case.\"},{\"question\":\"What are the main algorithmic improvements reported?\",\"answer\":\"It introduces a faster Monte Carlo algorithm for ESS with O*(1.6994^n) time and O*(1.5664^n) space, and also improves the best known polynomial-space running time to O*(2.5430^n).\"},{\"question\":\"How does the document handle correctness and randomness?\",\"answer\":\"The algorithms are Monte Carlo with no false positives, and their error probabilities can be reduced to 2^{-Ω(n)} by repeating the procedure.\"},{\"question\":\"What additional contribution is made beyond running-time improvements?\",\"answer\":\"The work investigates time-space tradeoffs for ESS and improves the achievable running times under various exponential-space 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