[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84646-en":3,"doc-seo-84646-105":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},84646,3848291630094,"Emma Wilson","https://eur-avatar.wpscdn.com/davatar_085a072bc5b1113ac321206ff7593b45",8,"Research & Report","Improved Approximation Algorithms for n-Pairs Shortest Paths","The paper studies the n-Pairs Shortest Paths problem, where a graph’s vertex set V defines n=|V| and distances must be approximated only for all n prespecified vertex pairs. It develops the first (2−α)^k-approximation algorithm for weighted undirected graphs, achieving a 1.622^k-approximation in O(mn^{1/k}+n^{1+2/k}) time. Results improve prior bounds for non-super-sparse graphs and extend to unweighted and dense weighted settings. The core technique is a new heavy-edge method that removes dependence on the maximum edge weight along shortest paths.","arXiv :2607 .02443v 1 [ cs .DS] 2 Jul 2026  \nImproved Approximation Algorithms for n-Pairs Shortest Paths  \nAvi Kadria, Liam Roditty, Virginia Vassilevska Williams  \nJuly 3, 2026  \nAbstract  \nLet G = (V, E) be a graph with n = |V | nodes and m = |E| edges. The t-Pairs Shortest Paths problem, introduced by Cohen [FOCS’93; SICOMP’99], asks to approximate the distances between tprespecified pairs of vertices. Recently, this problem has received renewed attention, particularly in the case where t = Θ(n): the n-Pairs Shortest Paths problem. In this setting, new algorithms and conditional lower bounds have been developed by Dalirrooyfard, Jin, Vassilevska Williams, and Wein [FOCS’22], and Chechik, Hoch, and Lifshitz [SODA’25] .  \nIn this paper, we present the first algorithm for the n-Pairs Shortest Paths problem in˜ weighted  \nundirected graphs that achieves a (2 − α)k-approximation, for constant α > 0, that runs in O (mn1/k + n1+2/k) time. Specifically, we present a 1 .622k-approximation, improving upon the (2k−3)-approximation of Chechik, Hoch, and Lifshitz [SODA’25] for graphs that are not super sparse, which answers in the affirmative the open question posed by them. We also develop improved approximation algorithms with better tradeoffs for unweighted graphs and dense weighted graphs that improve upon the results of Dalirrooyfard et al. and Chechik, Hoch, and Lifshitz.  \nOur main technical contribution is the new heavy-edge technique. Using this technique, we transform an algorithm with an approximation guarantee that depends on Wuv , the weight of the heaviest edge on the shortest path between u and v, into an algorithm with purely multiplicative approximation that does not depend on Wuv .  \n1 Introduction  \nLet G = (V, E) be a graph with n = |V | nodes and m = |E| edges, and let d (u, v) be the distance between u and v, for every u, v ∈ V. Let I ⊆ V × V be a set of t prespecified pairs of vertices. In the t-Pairs Shortest Paths (t-PSP) problem, the goal is to efficiently compute (or approximate) the shortest path distance d(u, v) for each pair (u, v) ∈ I. The t-PSP problem was first studied in the 1990s by Cohen [Coh98] and then by Aingworth, Chekuri, Indyk, and Motwani [ACIM99] . Aingworth et al. [ACI˜M99] presented an algorithm  \nthat computes a (1 , 2)-approximation 1 for unweighted, undirected graphs in O(n1.5 √t + n2 ) time. 2 A natural approach for approximating the t-PSP problem is to use the celebrated Approximate Distance  \nOra˜cles (ADOs) of Thoru˜ p and Zwick [TZ05] . These oracles are constructed for weighted undirected graphs  \nin O(mn1/k) time, use O (n1+1/k) space, and guarantee a (2k − 1)-approximation for distances between any pair of nodes. Once built, they can answer each˜ distance query in O (k) time, so solving the t-PSP problem  \nusing an ADO is straightforward and requires O(mn1/k + t) total time.  \nADOs are primarily designed to optimize space efficiency while supporting distance queries for all pairs of vertices. In contrast, the t-PSP problem only requires distance queries between a specific set of t vertex pairs, and the main objective is to minimize running time, not space. This distinction suggests that the generality of ADOs may be unnecessary in the t-PSP setting, and that one can hope to design faster algorithms that achieve improved time-approximation tradeoffs for t-PSP.  \nInterestingly, although the t-PSP problem was first introduced in the 1990s, it remained largely unexplored for over two decades. In the meantime, closely related problems, primarily focused on space-efficient representations such as pairwise spanners, distance preservers, and reachability preservers, received significantly more attention (see, for example,[CE06, AB18, KP22]) .  \nRecently, the t-PSP problem received renewed attention in the field of fine-grained complexity. Abboud et al. [ABKZ22, ABF23] and independently Jin and Xu [JX23] established several conditional lower bounds for t-PSP based on the 3SUM hypothesis. ","cbCaiaLud6l48GGe","https://ap.wps.com/l/cbCaiaLud6l48GGe","pdf",596464,1,22,"English","en",105,"# Abstract\n# Introduction\n## t-Pairs Shortest Paths background\n## Approximate Distance Oracles approach\n## Related lower bounds and prior work\n## New question and motivation","[{\"question\":\"What problem does the paper address?\",\"answer\":\"It addresses the n-Pairs Shortest Paths (n-PSP) problem, which requires approximating shortest-path distances only for a set of n prespecified vertex pairs in a graph.\"},{\"question\":\"What is the main approximation guarantee and running time?\",\"answer\":\"For weighted undirected graphs, it gives a first (2−α)^k-approximation algorithm, specifically achieving a 1.622^k-approximation in O(mn^{1/k}+n^{1+2/k}) time.\"},{\"question\":\"What key technical idea enables the improvement?\",\"answer\":\"The paper’s main technical contribution is a new heavy-edge technique that transforms an approximation depending on the heaviest edge weight on a shortest path into a purely multiplicative approximation independent of that 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problem does the paper address?","Question",{"text":75,"@type":76},"It addresses the n-Pairs Shortest Paths (n-PSP) problem, which requires approximating shortest-path distances only for a set of n prespecified vertex pairs in a graph.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What is the main approximation guarantee and running time?",{"text":80,"@type":76},"For weighted undirected graphs, it gives a first (2−α)^k-approximation algorithm, specifically achieving a 1.622^k-approximation in O(mn^{1/k}+n^{1+2/k}) time.",{"name":82,"@type":73,"acceptedAnswer":83},"What key technical idea enables the improvement?",{"text":84,"@type":76},"The paper’s main technical contribution is a new heavy-edge technique that transforms an approximation depending on the heaviest edge weight on a shortest path into a purely multiplicative approximation independent of that 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