[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82431-en":3,"doc-seo-82431-105":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},82431,7971461741311,"Ophelia","https://ap-avatar.wpscdn.com/avatar/74000253aff267980c6?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779345379180704826",8,"Research & Report","New Complexity Classes in Locally Checkable Labeling for Local Computation Algorithms","Local Computation Algorithms (LCAs) provide consistent query answers to large graph instances without storing state between queries. This paper studies LCA complexity via the classification question: for a target function f(n), which complexities are achievable up to polylogarithmic factors? The work focuses on Locally Checkable Labeling (LCL) problems, viewed as constant-degree constraint satisfaction tasks. New LCL constructions are presented in the VOLUME model, and extended to LCAs, yielding probe complexities Θ(log^k n) and Θ(n^(p/q)) for k≥1 and rational p/q in (0,1].","arXiv :2607 .09626v1 [ cs .DC] 10 Jul 2026  \nNew Complexity Classes in Locally Checkable Labeling for Local Computation Algorithms  \nSijin Peng \\# 􀀚  \nCSAIL, MIT, Cambridge, United States  \n~~ Abstract ~~  \nLocal Computation Algorithms (LCAs), introduced by Rubinfeld, Tamir, Vardi, and Xie (2011), area special type of sublinear algorithms that, given probing access to a possibly massive input, are required to provide query access to a consistent solution, without maintaining a state between different queries. In this paper, we try to understand LCA through the lens of complexity classifications, described by the following question: Given a target complexity function f (n), is there a problem whose local computation complexity is f (n), up to polylogarithmic factors?  \nWe restrict our focus to Locally Checkable Labeling (LCL) problems, which can be seen as constant-degree constraint satisfaction problems. Possible complexity classes of this problem family have been extensively studied in various distributed computation models, including the VOLUME model proposed by Rosenbaum and Suomela (2020), which is an invariant of local computation algorithms with additional locality requirements.  \nIn this paper, we provide new LCL complexity constructions in the VOLUME model, and generalize the results to LCAs. Specifically, we show tha˜t there are LCLs whose probe complexities  \nin the VOLUME and LCA models are Θ(logk n) and Θ (np/q ) for any positive integer k ≥ 1 and rational p/q ∈ (0 , 1] . Our approach, completely different from the approach to a similar result in the d˜istributed LOCAL model by Balliu et al. (2018), is to stack instances of complexity Θ(log n) and  \nΘ(n1/k ) in the VOLUME model constructed by Rosenbaum and Suomela (2020) .  \n2012 ACM Subject Classification Theory of computation → Distributed algorithms; Theory of computation → Streaming, sublinear and near linear time algorithms  \nKeywords and phrases Local Computation Algorithms, Volume Model, Locally Checkable Labeling  \nAcknowledgements The LCL construction, lower bound analysis, and high-level ideas for upper bound algorithms in this paper were discovered by the author. At the same time, most of the proof details in Sections 4 and 5, including Definition 40, and figures in Sections 1 and 2 were assisted by the use of OpenAI Codex. All AI outputs were reviewed and edited by the author, who takes full responsibility for the correctness, originality, and integrity of this paper.  \n 1  Introduction  \nLocal computation algorithms (LCAs), proposed in [2, 42], are a special type of sublinear algorithms typically applied to graph problems involving multiple output bits. An LCA is expected to find only a small part of the solution each time, specified by queries, through a small number of probes that give the algorithm access to the input and adjacency list of a vertex in the graph. Following most of the LCA literature, in this paper, the input graph has a constant maximum degree, so it is the same up to a constant factor for a probe to return the complete adjacency list of a vertex as to return one entry of the adjacency list.  \nThe main challenge in designing a local computation algorithm is consistency: when there are multiple feasible solutions to the instance, multiple queries to different parts of the solution should provide consistent local solutions that point to a single globally feasible solution without maintaining state across queries. For graph coloring, for example, a query may ask for the color of one vertex, and the answers to multiple queries must together forma single feasible coloring. To make this possible, if an LCA needs randomness, the same randomness must be shared across queries.  \n2 New LCL Complexity Classes for LCAs  \nThe motivation behind LCA is to provide an efficient and parallelizable way to gain access to a small part of a large solution to a massive problem. This shares the spirit with distributed computing and property testing. See [35","cbCaie4F1X7FKLNf","https://ap.wps.com/l/cbCaie4F1X7FKLNf","pdf",788916,1,46,"English","en",105,"# Introduction\n# New LCL Complexity Classes for LCAs","[{\"question\":\"What problem does the paper address in local computation algorithms?\",\"answer\":\"The paper studies which probe complexity functions f(n) can be realized by local computation algorithms up to polylogarithmic factors, using a complexity classification lens rather than focusing only on specific optimal problems.\"},{\"question\":\"What are Locally Checkable Labeling (LCL) problems in this work?\",\"answer\":\"LCL problems are treated as constant-degree constraint satisfaction tasks defined on bounded-degree graph families, where local labels must satisfy constraints that are checkable within a constant neighborhood.\"},{\"question\":\"What probe complexity results are obtained in the VOLUME model and for LCAs?\",\"answer\":\"The paper constructs LCL instances whose probe complexities in the VOLUME and LCA models are Θ(log^k n) and Θ(n^(p/q)), respectively, for any integer k≥1 and rational p/q in 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problem does the paper address in local computation algorithms?","Question",{"text":75,"@type":76},"The paper studies which probe complexity functions f(n) can be realized by local computation algorithms up to polylogarithmic factors, using a complexity classification lens rather than focusing only on specific optimal problems.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What are Locally Checkable Labeling (LCL) problems in this work?",{"text":80,"@type":76},"LCL problems are treated as constant-degree constraint satisfaction tasks defined on bounded-degree graph families, where local labels must satisfy constraints that are checkable within a constant neighborhood.",{"name":82,"@type":73,"acceptedAnswer":83},"What probe complexity results are obtained in the VOLUME model and for LCAs?",{"text":84,"@type":76},"The paper constructs LCL instances whose probe complexities in the VOLUME and LCA models are Θ(log^k n) and Θ(n^(p/q)), respectively, for any integer k≥1 and rational 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