[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85016-en":3,"doc-seo-85016-105":29,"detail-sidebar-cat-0-en-105":90},{"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":4,"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},85016,1374391975076,"Riley","https://ap-avatar.wpscdn.com/avatar/14000253ca4ec9f6853?x-image-process=image/resize,m_fixed,w_180,h_180&k=1783305029341752051",8,"Research & Report","Exploiting Spanning Trees for Directed Acyclicity","Weighted Maximum Acyclic Subgraph (MAS) is studied for directed graphs with positive edge weights, aiming to find a maximum-weight acyclic edge set. The work contrasts the random ordering half-weight guarantee with a new MaxST-based lower bound derived from a maximum-weight acyclic subgraph of the underlying undirected graph. Two parameterized algorithms are presented that either find an acyclic set with weight at least MaxST(G)+k or report none, one for integral weights in time 2kO(1)·|I|O(1) and one for rational weights in time nkO(1)·|I|O(1).","Exploiting Spanning Trees for Directed Acyclicity ∗†  \nSergei Khargeliia‡ [sergey.khargelia@gmail.com](sergey.khargelia@gmail.com)  \nDanil Sagunov§ [danilka.pro@gmail.com](danilka.pro@gmail.com)  \narXiv :2607 .07705v 1 [ cs .DS] 8 Jul 2026  \nAbstract  \nWe study the weighted case of the Maximum Acyclic Subgraph (MAS) problem, where each edge of a given directed graph has a positive weight assigned, and the task is to find a maximum-weight acyclic edge set. The famous and well-studied random ordering lower bound guarantees the existence of an acyclic set that gives at least the half of the total edge weight.  \nThe maximum spanning tree (MaxST) guarantee, which is the weight of a maximum-weight acyclic subgraph of the underlying undirected graph of G, is another natural lower bound for the weight of an acyclic subgraph. A solution of this weight dominates the random ordering solution on instances where MaxST spans the most of the total edge weight.  \nOur main contribution are two parameterized algorithms that find acyclic subgraphs of total weight larger than the weight of the MaxST of G. Both our algorithms find a solution of total weight at least MaxST(G) + k, for a given integer k ≥ 0, or report that it does not exist, and  \n• First of our algorithms runs in time 2kO(1) ·|I| O(1) and works when all weights are integers;  \n• Our second algorithm handles rational weights not less than 1, and its running time is upper-bounded by nkO(1) · |I| O(1) . This positive result is rather surprising since solving MAS above the random ordering lower bound is NP-hard in the same rational weights scenario, when k = 1 .  \nOur findings unravel intricate connections between structure of MaxSTs and directed cycles, use perfect graph theorem to tackle rational weights, and raise graph-theoretic questions that are interesting on their own. Of another importance, this is one of the few examples of positive“above guarantee” results for a weighted problem on directed graphs, especially for rational weights.  \n∗ Sergei Khargeliia was supported by PJSC «Gazprom Neft», aggr. \\# ГПН-26/03000/00502/Р .  \n†Danil Sagunov was supported by the Ministry of Science and Higher Education of the Russian Federation (agreement 075-15-2025-344 dated 29/04/2025 for Saint Petersburg Leonhard Euler International Mathematical Institute at PDMI RAS) .  \n‡ITMO University, Saint Petersburg State University, St. Petersburg, Russia.  \n§ Markov Lab, Saint Petersburg State University, St. Petersburg, Russia.  \nContents  \n1 Introduction 3  \n2 Preliminaries 6  \n3 Classification of directed edges and their basic properties 6  \n3.1 Maximum spanning tree, allowed and blocked edges ................... 7  \n3.2 Inverse edges and their properties ............................. 7  \n3.3 Tree edge profits ...................................... 9  \n3.4 Remaining profits ...................................... 10  \n4 FPT-algorithm for integral edge weights 10  \n4.1 Limiting allowed edges ................................... 10  \n4.2 Decomposing MST into paths ............................... 11  \n4.3 Greedy-like approach to remaining profits ........................ 17  \n4.4 Solution when the removed tree edges are fixed ..................... 21  \n4.5 Putting all together ..................................... 23  \n5 XP-algorithm for rational edge weights 25  \n5.1 Maximizing profit under restrictions ........................... 25  \n5.2 Dealing with allowed edges ................................ 31  \n5.2.1 Inv-respecting subgraphs .............................. 32  \n5.2.2 Compressed representation of acyclic subgraphs ................. 32  \n5.2.3 Forbidding paths via constraint sets ....................... 33  \n5.3 Summing up ......................................... 43  \n6 Conclusion 44  \n1 Introduction  \nIn the Maximum Acyclic Subgraph problem, MAS for short, we are given a directed graph (digraph) G with n vertices and m edges, and an integer k ≥ 0, and the task is to find an acyclic subgraph o","cbCaigC4BYU7a5kr","https://ap.wps.com/l/cbCaigC4BYU7a5kr","pdf",836665,1,47,"English","en",105,"# Introduction\n# Preliminaries\n# Classification of directed edges and their basic properties\n## Maximum spanning tree, allowed and blocked edges\n## Inverse edges and their properties\n## Tree edge profits\n## Remaining profits\n# FPT-algorithm for integral edge weights\n## Limiting allowed edges\n## Decomposing MST into paths\n## Greedy-like approach to remaining profits\n## Solution when the removed tree edges are fixed\n## Putting all together\n# XP-algorithm for rational edge weights\n## Maximizing profit under restrictions\n## Dealing with allowed edges\n## Summing up\n# Conclusion","[{\"question\":\"What problem does the paper study in directed graphs?\",\"answer\":\"The paper studies the weighted Maximum Acyclic Subgraph (MAS) problem, where each directed edge has a positive weight and the goal is to select an acyclic edge set with maximum total weight.\"},{\"question\":\"How does the MaxST guarantee relate to the random ordering lower bound?\",\"answer\":\"The MaxST guarantee is a natural lower bound based on the weight of a maximum-weight acyclic subgraph of the underlying undirected graph. The paper states that this bound can dominate the random ordering solution on instances where the MaxST spans most of the total edge weight.\"},{\"question\":\"What do the two main algorithms guarantee about acyclic subgraphs?\",\"answer\":\"Both parameterized algorithms aim to find an acyclic subgraph whose total weight is at least MaxST(G)+k for any integer k≥0, or report that such a solution does not exist. The first handles integral weights, while the second handles rational weights and is based on additional structural tools.\"}]",1784200311,118,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":85,"head_meta":87,"extra_data":89,"updated_unix":27},"exploiting-spanning-trees-for-directed-acyclicity","",{"@graph":35,"@context":84},[36,53,67],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/exploiting-spanning-trees-for-directed-acyclicity/85016/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":61,"encodingFormat":60,"isAccessibleForFree":62,"interactionStatistic":63},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-16",true,{"@type":64,"interactionType":65,"userInteractionCount":4},"InteractionCounter",{"@type":66},"ViewAction",{"@type":68,"mainEntity":69},"FAQPage",[70,76,80],{"name":71,"@type":72,"acceptedAnswer":73},"What problem does the paper study in directed graphs?","Question",{"text":74,"@type":75},"The paper studies the weighted Maximum Acyclic Subgraph (MAS) problem, where each directed edge has a positive weight and the goal is to select an acyclic edge set with maximum total weight.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"How does the MaxST guarantee relate to the random ordering lower bound?",{"text":79,"@type":75},"The MaxST guarantee is a natural lower bound based on the weight of a maximum-weight acyclic subgraph of the underlying undirected graph. The paper states that this bound can dominate the random ordering solution on instances where the MaxST spans most of the total edge weight.",{"name":81,"@type":72,"acceptedAnswer":82},"What do the two main algorithms guarantee about acyclic subgraphs?",{"text":83,"@type":75},"Both parameterized algorithms aim to find an acyclic subgraph whose total weight is at least MaxST(G)+k for any integer k≥0, or report that such a solution does not exist. The first handles integral weights, while the second handles rational weights and is based on additional structural tools.","https://schema.org",{"og:url":51,"og:type":86,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":88,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":91},[92,96,100,104,109,114,119,122,127,130,134],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":93,"show_sort_weight":94,"slug":95},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":97,"show_sort_weight":98,"slug":99},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":101,"show_sort_weight":102,"slug":103},"Exam",70,"exam",{"id":105,"doc_module":4,"doc_module_name":45,"category_name":106,"show_sort_weight":107,"slug":108},5,"Comic",60,"comic",{"id":110,"doc_module":4,"doc_module_name":45,"category_name":111,"show_sort_weight":112,"slug":113},6,"Technology",50,"technology",{"id":115,"doc_module":4,"doc_module_name":45,"category_name":116,"show_sort_weight":117,"slug":118},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":45,"category_name":124,"show_sort_weight":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":45,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":45,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":45,"category_name":136,"show_sort_weight":105,"slug":137},19,"General","general"]