[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83544-en":3,"doc-seo-83544-105":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},83544,962075006959,"Anda","https://ap-avatar.wpscdn.com/avatar/e0002397efbe92a78e?_k=1776741047341049297",8,"Research & Report","Warm-Starting All-Pairs Shortest Paths with Predictions","One of the key hypotheses in fine-grained complexity posits that All-Pairs Shortest Paths (APSP) needs cubic time, up to subpolynomial factors. This work studies APSP in the learning-augmented setting, where an algorithm receives predictions derived from similar past instances. It uses vertex sets indicating the shortest detour for each pair and achieves runtime O(n^2.83 + ηn), with η measuring prediction insufficiency. The approach leverages a co-nondeterministic Exact Triangle method to validate and recover from nondeterministic-certificate errors. These results form a first step toward learning-augmented algorithms under APSP-conditioned lower bounds.","arXiv :2607 .00857v 1 [ cs .DS] 1 Jul 2026  \nWarm-Starting All-Pairs Shortest Paths with Predictions  \nAdam Polak \\# Ñ 􀀚  \nBocconi University, Milan, Italy Jonas Schmidt \\# Ñ 􀀚 Bocconi University, Milan, Italy  \n~~ Abstract ~~  \nOne of the three key hypotheses of fine-grained complexity asserts that computing All-Pairs Shortest Paths (APSP) requires cubic time, up to subpolynomial factors, in the worst case. We initiate the study of APSP in the paradigm of algorithms with predictions, also known as learning-augmented algorithms. We propose an APSP algorithm that takes as additional input a prediction (e.g., given by a model learned from similar instances seen in the past) consisting of sets of vertices causing the shortest detour for each pair of vertices. The algorithm runs in time O (n2 .83 + ηn), where η denotes the prediction error defined as the number of pairs of vertices for which, informally speaking, the prediction was not sufficient to compute and certify optimality of the shortest path length. This is already subcubic when the prediction error is (polynomially) smaller than its maximum possible values n2 , i.e. , whenever the prediction is at least slightly better than terrible.  \nWe build on the co-nondeterministic algorithm for the Exact Triangle problem by Chan, Vassilevska Williams, and Xu (STOC 2023), essentially enabling this algorithm to detect mistakes in the nondeterministic certificate and recover from them.  \nOur result constitutes the first necessary step towards designing learning-augmented algorithms for problems with known fine-grained lower bounds conditioned on the APSP Hypothesis.  \nSupplementary Material [https://github.com/adampolak/warm-starting-apsp](https://github.com/adampolak/warm-starting-apsp)  \nFunding Funded by the Italian Ministry of University and Research (MUR) under Ministerial Decree No. 18169 of 17 November 2025 – FIS 3 Call CUP: J53C25002160001  \nAcknowledgements We thank Antonios Antoniadis and Yasamin Nazari for helpful discussion.  \n 1  Introduction  \nGiven a directed edge-weighted graph G = (V, E, w), the All-Pairs Shortest Paths (APSP) problem asks to compute for each pair of vertices u, v ∈ V the distance from u to v, i.e. , the minimum total weight of edges on a path from u to v. In n-vertex graphs, APSP can be easily solved in time O (n3 ) , e.g. , using the Floyd–Warshall algorithm [37, 23 , 44], or, when the edge weights are non-negative, by running Dijkstra’s algorithm [18] from each vertex. The fastest known APSP algorithm [45] shaves off only a factor of 2O ( √log n) , and the APSP Hypothesis [41] asserts that the cubic running time is optimal up to subpolynomial factors. It is one of the three main hypotheses of fine-grained complexity [40], alongside the 3SUM Hypothesis and Strong Exponential Time Hypothesis (SETH) .  \nImagine we solve many instances of APSP that are in some sense similar to each other, e.g., many snapshots of a road network taken over time with edge weights representing estimated travel times that get updated between the snapshots.  \nCan we use a common structure shared by similar APSP instancesin order to solve them faster?  \nThis question fits into the recent line of research on learning-augmented algorithms, also known as algorithms with predictions, see, e.g., [35, 32] . In particular, it is the kind  \n2 Warm-Starting All-Pairs Shortest Paths with Predictions  \nof question asked in papers on warm-starting algorithms (e.g., [19, 38 , 7]) . There, the algorithm’s input is enriched with a hint or prediction, produced, e.g., by a model trained on previously solved similar instances. The algorithm must remain correct unconditionally, but its running time is allowed to depend smoothly on the quality of the prediction – quantified by an error measure denoted by η . In particular, even a mildly informative prediction should already yield a provable speedup.  \nMany warm-starting results to date address problems that already admit algorithms running in a","cbCait8SywXYWBN4","https://ap.wps.com/l/cbCait8SywXYWBN4","pdf",807123,3,1,20,"English","en",105,"# Introduction\n## Algorithms with Predictions (Learning-Augmented)\n## Fine-Grained Complexity Background\n## Warm-Starting and Similarity Notions","[{\"question\":\"What problem does the paper address?\",\"answer\":\"The paper addresses the All-Pairs Shortest Paths (APSP) problem: computing distances between every pair of vertices in a directed weighted graph.\"},{\"question\":\"How does the proposed approach use predictions?\",\"answer\":\"It enriches APSP with an additional prediction specifying, for each vertex pair, sets of vertices that cause the shortest detour, typically obtained from a model trained on similar past instances.\"},{\"question\":\"What does the runtime bound depend on?\",\"answer\":\"The runtime is O(n^2.83 + ηn), where η is the prediction error measuring how many vertex pairs the prediction fails to support for certifying optimality of the shortest-path 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problem does the paper address?","Question",{"text":75,"@type":76},"The paper addresses the All-Pairs Shortest Paths (APSP) problem: computing distances between every pair of vertices in a directed weighted graph.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the proposed approach use predictions?",{"text":80,"@type":76},"It enriches APSP with an additional prediction specifying, for each vertex pair, sets of vertices that cause the shortest detour, typically obtained from a model trained on similar past instances.",{"name":82,"@type":73,"acceptedAnswer":83},"What does the runtime bound depend on?",{"text":84,"@type":76},"The runtime is O(n^2.83 + ηn), where η is the prediction error measuring how many vertex pairs the prediction fails to support for certifying optimality of the shortest-path 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