[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85806-en":3,"doc-seo-85806-105":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},85806,8796095462418,"Noah","https://ap-avatar.wpscdn.com/avatar/80000253c1241d02b47?x-image-process=image/resize,m_fixed,w_180,h_180&k=1778826106357471780",8,"Research & Report","Distributed Load Balancing on Unrelated Machines","Distributed load balancing is studied in the distributed CONGEST communication model using bipartite graphs where jobs must be assigned to unrelated machines with arbitrary nonnegative job sizes sij per machine–job pair. The objective minimizes the maximum machine load. Earlier CONGEST results produced (1+ε)-approximate fractional solutions only for restricted edge sizes and relied on rounding to obtain a best-known (2+ε)-approximate integral solution. A new CONGEST algorithm handles general sizes sij while preserving polylog round optimality.","arXiv :2607 . 10075v 1 [ cs .DS] 11 Jul 2026  \nDistributed Load Balancing on Unrelated Machines Aaron Bernstein∗ Anupam Gupta† Zhaozi Wang†  \nJuly 2026  \nAbstract  \nWe study the well-known load balancing problem in the distributed (CONGEST) model of computation. The input is a bipartite graph G = (J ∪ M, E), with jobs J to be assigned to machines M. We consider the unrelated machines setting, where the input specifies an arbitrary non-negative size sij for every machine i and job j. The goal is to find an assignment φ : J → M that minimizes the maximum machine load, where the load of a machine i is Pj:φ(j)=i sij , i.e. , the total size of the jobs assigned to it.  \nIn the CONGEST model, the state-of-the-art is an algorithm that runs in polylog roundsand returns a (1 + ε)-approximate fractional solution [ALPZ21] . This can be combined with a rounding scheme to also yield a (2 + ε)-approximate integral solution, which is the best polynomial-time result known even in the centralized setting. However, this algorithm, as well as all previous CONGEST algorithms, can only solve a special case of load balancing, where all edges incident to a job j have the same size sj .  \nOur main contribution is a CONGEST algorithm that can handle the case of general sizes sij . The algorithm is again essentially optimal: it computes a (1+ε)-approximate fractional solution or a (2 + ε)-approximate integral solution in polylog rounds. The problem structure changes significantly once we allow arbitrary edge-sizes, so our techniques are very different from those used in previous algorithms for distributed load balancing.  \nOne ingredient of our result is a black-box tool that is of independent interest: a (1 + ε)-approximation algorithm to arbitrary mixed packing-covering linear programs in the CONGEST model in polylog rounds. While such an algorithm was already known in the more powerful parallel model, previous polylog-round algorithms in the distributed CONGEST model only solved pure packing or pure covering problems. We improve upon a very recent CONGEST algorithm for mixed packing-covering that runs in O(D polylog) rounds, where D is the diameter of the corresponding communication graph.  \n∗ New York University. Supported by Sloan Fellowship, Google Research Fellowship, NSF Grant 1942010, and Charles S. Baylis endowment at NYU.  \n†Department of Computer Science, Courant Institute of Mathematical Sciences, New York University. Supported in part by NSF awards CCF-2422926 and CCF-2608359 .  \n1 Introduction  \nIn this work we study the load balancing problem in the distributed (CONGEST) model of computation. The input is a bipartite graph G = (J ∪ M, E), where we refer to the sets J and M asthe jobs and machines, respectively. We consider the unrelated machines model, where the size of each job j on a machine i is denoted sij, and these sizes can be completely unrelated to each other.(If there is no edge (i, j) ∈ E, then we imagine that sij = ∞ ; i.e., assigning job j to machine i is forbidden.) The goal is to find an assignment φ : J → M that minimizes the maximum machine load, where the load of a machine i is Pj:φ(j)=isij, i.e., the total size of jobs assigned to it.  \nThere is a rich literature on load balancing, both within the graph algorithms community (where it is sometimes called semi-matching), and the scheduling community (where it is called makespan minimization) . It has been studied within many different models of computation, from classical polynomial-time approximation algorithms [LST90, ST93], to fast algorithms [HLLT06, FLN14], online algorithms (both in the classical model [ANR95, AAF+97], as well as in settings with recourse [GKS14, KLS23] and other beyond-worst-case scenarios [AFGS22, GM26 , LX21 , IKL+24]), streaming algorithms [KR16, ABL23 , ABL+25], parallel algorithms [You01, MRWZ16 , Li23], and dynamic algorithms [BKS23] . In the standard centralized model, one can solve the fractional version exactly via linear programming, an","cbCaimdfhLPMq28V","https://ap.wps.com/l/cbCaimdfhLPMq28V","pdf",631758,1,33,"English","en",105,"# Abstract\n# Introduction\n## Distributed Model\n## Distributed Load Balancing\n## Distributed Mixed Packing-Covering LPs\n## Main Contribution","[{\"question\":\"What problem does the document address in the CONGEST model?\",\"answer\":\"It addresses distributed load balancing on unrelated machines, where jobs are assigned to machines to minimize the maximum machine load in the CONGEST communication setting.\"},{\"question\":\"What limitation do earlier CONGEST algorithms have?\",\"answer\":\"Earlier CONGEST algorithms could only solve a special restricted case where all edges incident to a job share the same size sj.\"},{\"question\":\"What is the key contribution for the general unrelated edge sizes sij?\",\"answer\":\"The document presents a CONGEST algorithm that computes a (1+ε)-approximate fractional solution or a (2+ε)-approximate integral solution in polylog rounds, using techniques tailored to arbitrary edge sizes and a mixed packing-covering LP 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problem does the document address in the CONGEST model?","Question",{"text":75,"@type":76},"It addresses distributed load balancing on unrelated machines, where jobs are assigned to machines to minimize the maximum machine load in the CONGEST communication setting.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What limitation do earlier CONGEST algorithms have?",{"text":80,"@type":76},"Earlier CONGEST algorithms could only solve a special restricted case where all edges incident to a job share the same size sj.",{"name":82,"@type":73,"acceptedAnswer":83},"What is the key contribution for the general unrelated edge sizes sij?",{"text":84,"@type":76},"The document presents a CONGEST algorithm that computes a (1+ε)-approximate fractional solution or a (2+ε)-approximate integral solution in polylog rounds, using techniques tailored to arbitrary edge sizes and a mixed packing-covering LP 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