[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85619-en":3,"doc-seo-85619-105":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},85619,4810365810221,"Aurora","https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d",8,"Research & Report","A Note On Rounding Fractional Matchings With Constant-Factor Strong Negative Correlation","We present new dependent-rounding algorithms for bipartite graphs that convert a fractional matching x into an integral selection X. Each right node v in V is incident to exactly one chosen edge, while variables Xe exhibit broad non-positive correlation. For any two distinct edges e and f sharing a left node u, the algorithm guarantees strong negative correlation, with E[XeXf] far below xexf. The work yields simpler, stronger bounds than prior approaches, including an improved constant 0.79751 and a tight lower limit of 1/2.","arXiv :2606 .07820v2 [ cs .DS] 11 Jul 2026  \nA note on rounding fractional matchings with constant-factor strong negative correlation  \nDavid G. Harris∗  \nJuly 14, 2026  \nAbstract  \nWe describe new dependent-rounding algorithms for bipartite graphs. Given a fractional matching x of graph G = (U∪V, E), the algorithms return an integral solution X such that each right-node v ∈ V has exactly one edge, and where the variables Xe also satisfy broad non-positive correlation properties. In particular, for distinct edges e, f sharing a left-node u ∈ U, the variables Xe , Xf have strong negative-correlation, i.e. the expectation of XeXf is significantly below xexf .  \nDependent rounding schemes with these properties have been used for approximation algorithms for job-scheduling on unrelated machines to minimize weighted completion times, among other applications. Our new algorithm achieves simpler and qualitatively stronger bounds compared to prior algorithms. In particular, we achieve a negative-correlation property  \nE [XeXf] ≤ 0.79751 xexf ,  \nwhich is a significant constant-factor improvement over Baveja, Qu & Srinivasan (2024) .  \nWe show that the constant cannot be reduced below 1/2 by any comparable rounding algorithm.  \n1 Introduction  \nMany scheduling and resource allocation problems can be formulated in terms of a bipartite assignment problem: we are given a complete bipartite graph G = (U ∪ V, E), and we wish to select a “half-matching” K, i.e. a set of edges K which intersects each right-node v ∈ V exactly once. For instance, V can represent a set of jobs to be scheduled, and U can represent a set of possible machines. Alternatively, V can represent items to be sold, and U can represent potential buyers.  \nA popular strategy for such problems is to first solve a relaxation (e.g. , a linear program), obtaining fractional vectors xe : e ∈ E, and then round this to an integral solution Xe : e ∈ E where Xe is the indicator that e ∈ K. For this strategy to be viable, the fractional solution x must satisfy x(N(v)) = 1 for all right-nodes v. (Here, N (v) denotes the set of edges incident on vertex vand we write x(L) = Pe∈L xe for any edge-set L) . One particularly nice scenario has the left-nodes also satisfying x(N(u)) = 1, i.e. a fractional matching.  \nThe simplest rounding method, known as independent rounding, is that each right-node v ∈ V independently selects one neighboring edge e ∈ N (v), wherein each edge is selected with probability xe. However, the left-nodes (e.g. the machines in a scheduling problem) can then become unevenly loaded due to random fluctuations. This can be undesirable for some allocation problems.  \nA new rounding approach was proposed in [BSS21], based on dependent rounding with strong negative correlation. This was applied to a classical scheduling problem of minimizing weighted completion time on unrelated machines. The edges are tied together in a scheme wherein each edge e is still marginally selected with probability xe , while neighboring edge pairs satisfy a stronger property:  \nE [XeXf] ≤ c xexf for a constant c \u003C 1 (1)  \nOf course, independent rounding would give E[XeXf ] = xexf exactly.  \n∗ University of Maryland. Email: [davidgharris29@gmail.com](davidgharris29@gmail.com)  \nThe guarantee of (1) is known as (constant-factor) strong negative correlation. Since then, a variety of rounding schemes with strong negative correlation have been developed, leading to improved approximation ratios for various scheduling problems [IS20, IL23, BQS24, Har25, HLRV26] . Generally speaking, these fall into two classes. The first, which includes the original work of [BSS21], uses a random walk: at each stage, the fractional vector x is modified, until eventually it becomes integral. The modification rule is chosen so that if two edges e, f share a left-node, then one value xe is incremented and the other value xf is decremented. An improved version of this rounding scheme was later developed in [BQS24] .  \nTh","cbCaigXpTkKdYq7H","https://ap.wps.com/l/cbCaigXpTkKdYq7H","pdf",294383,2,1,9,"English","en",105,"# Abstract\n# Introduction\n## Problem setup and motivation\n## Prior work and two rounding paradigms\n## Our contribution","[{\"question\":\"What does the dependent-rounding algorithm take as input and output?\",\"answer\":\"It takes a fractional matching x on a bipartite graph and outputs an integral edge selection X. The output ensures every right node in V is incident to exactly one chosen edge.\"},{\"question\":\"How is the strong negative correlation property stated?\",\"answer\":\"For two distinct edges e and f sharing a left node, the algorithm guarantees E[XeXf] is significantly below xexf, specifically giving E[XeXf] ≤ 0.79751 xexf.\"},{\"question\":\"Why does the paper claim the constant factor cannot be improved below 1/2?\",\"answer\":\"The paper shows that no comparable rounding algorithm can reduce the constant in the strong negative-correlation bound below 1/2.\"}]",1784204968,23,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"a-note-on-rounding-fractional-matchings-with-constant-factor-strong-negative-correlation","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,47,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":20},"https://docshare.wps.com/document/","Document",{"item":48,"name":12,"@type":43,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/a-note-on-rounding-fractional-matchings-with-constant-factor-strong-negative-correlation/85619/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-24","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What does the dependent-rounding algorithm take as input and output?","Question",{"text":75,"@type":76},"It takes a fractional matching x on a bipartite graph and outputs an integral edge selection X. The output ensures every right node in V is incident to exactly one chosen edge.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How is the strong negative correlation property stated?",{"text":80,"@type":76},"For two distinct edges e and f sharing a left node, the algorithm guarantees E[XeXf] is significantly below xexf, specifically giving E[XeXf] ≤ 0.79751 xexf.",{"name":82,"@type":73,"acceptedAnswer":83},"Why does the paper claim the constant factor cannot be improved below 1/2?",{"text":84,"@type":76},"The paper shows that no comparable rounding algorithm can reduce the constant in the strong negative-correlation bound below 1/2.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,127,130,134],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":22,"doc_module":4,"doc_module_name":46,"category_name":124,"show_sort_weight":125,"slug":126},"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":46,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":46,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":46,"category_name":136,"show_sort_weight":106,"slug":137},19,"General","general"]