[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85022-en":3,"doc-seo-85022-105":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},85022,1374391975076,"Riley","https://ap-avatar.wpscdn.com/avatar/14000253ca4ec9f6853?x-image-process=image/resize,m_fixed,w_180,h_180&k=1783305029341752051",8,"Research & Report","Domination and Coverage Problems under Vulnerability Constraints","Domination and coverage problems are studied with restrictions that mark certain vertices or edges as vulnerable and therefore not to be dominated or covered. The work defines k-Vertex Maximum Domination Ratio with Vulnerable Vertices (k-Max DRVV) and develops an approximation algorithm for bounded-degree graphs. A related Maximum Domination Ratio with Vulnerable Vertices (DRVV) variant supports the approximation. It introduces Dominating Set with Vulnerable Vertices (DSV) via reduction to Red-Blue Set Cover and derives a harmonic-based approximation bound, then formulates Vertex Cover with Vulnerable Edges (VCVE) and provides a tight polynomial-time 2-approximation.","arXiv :2607 .07842v 1 [ cs .DS] 8 Jul 2026  \nDomination and Coverage Problems under Vulnerability Constraints  \nIoannis Sigalas Nikolaos Lazaropoulos Ioannis Lamprou  \nIoannis Vaxevanakis Vassilis Zissimopoulos  \n1 Department of Informatics and Telecommunications, National and Kapodistrian University of Athens, Greece  \nIn various domination and coverage problems, certain vertices or edges should not be dominated/covered and are designated as vulnerable. Motivated by this, we define the k-Vertex Maximum Domination Ratio with Vulnerable Vertices (k-Max DRVV ) problem, which extends the budgeted dominating set problem to include vulnerability constraints. We propose an approximation algorithm based on an unbudgeted variant of k-Max DRVV , termed the Maximum Domination Ratio with Vulnerable Vertices (DRVV ) problem. For bounded-degree graphs of order n, our algorithm provides an O (k/n)-approximation for the k-Max DRVV problem. We introduce the Dominating Set with Vulnerable Vertices (DSV ) problem, reduce it to the Red-Blue Set Cover problem, and derive a 2 q |V | · (H(∆N ) − ~~1~~2 )-approximation algorithm, where |V | is the order of the graph, ∆N is the maximum degree among non-vulnerable vertices and H is the harmonic function. Finally, we examine the Vertex Cover with Vulnerable Edges ( VCVE) problem, which can be naturally expressed as a special case of the Red-Blue Set Cover problem. We present a polynomial-time 2-approximation algorithm for the V CVE problem, achieving the best possible ratio.  \nKeywords: Approximation Algorithms, Coverage, Influence Maximization, Vulnerable, Domination.  \n1 Introduction  \nDomination and Coverage problems have been studied extensively in the literature Haynes et al. (2013, 2020); Vazirani (2001) . In this work, we focus on variants of these problems with constraints regarding vulnerable vertices or edges. These formulations are closely connected to the classical Red-Blue Set Cover problem Carr et al. (2000); Peleg (2007) and are motivated by applications in influence maximization, where the diffusion cascade model is typically used to capture how ideas or behaviors spread in networks while respecting vulnerable elements. In contrast to cascade-based formulations, our perspective abstracts the diffusion process into structural graph properties, aiming to maximize the number of dominated vertices or covered edges while controlling the number of vulnerable ones.  \nInfluence maximization aims to select a set of seed vertices whose activation triggers the largest spread of ideas or behaviors throughout the network. There has been significant research Pasumarthi et al. (2015);  \nWen et al. (2018) focusing on optimizing the influence on a specific target audience. Socially responsible influence maximization Chen et al. (2021) further emphasizes minimizing influence on vulnerable vertices. A common metric to evaluate this trade-off is the Additively Smoothed Ratio (ASR), the fraction 1365–8050 © 2005 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France  \n2 I. Sigalas et al.  \nof expected influenced non-vulnerable vertices over vulnerable ones. We, on the other hand, focus on combinatorial formulations that represent vulnerability constraints directly on the graph.  \nBuilding on this perspective, we define the k-Vertex Maximum Domination Ratio with Vulnerable Vertices (k-Max DRVV ), which maximizes the ratio of dominated non-vulnerable over vulnerable vertices. This formulation offers a clear combinatorial abstraction of socially responsible influence maximization and provides a framework in which vulnerability constraints can be studied through domination and coverage problems. Vertex Cover with vulnerable edges (VCVE) is an example of coverage problem with vulnerable edges.  \nBeyond applications in social networks, the concept of Red-Blue vertices (or vulnerable and nonvulnerable vertices in our formulation) extends to a variety of other domains. For instance, data minin","cbCaikcco9CW183i","https://ap.wps.com/l/cbCaikcco9CW183i","pdf",469750,1,24,"English","en",105,"# Introduction\n## Related Work","[{\"question\":\"What is the motivation behind vulnerability constraints in domination and coverage problems?\",\"answer\":\"The framework marks certain vertices or edges as vulnerable so they must not be dominated or covered. This models socially responsible objectives, and is also tied to influence maximization perspectives on network diffusion while controlling the impact on vulnerable elements.\"},{\"question\":\"How is the k-Max DRVV problem defined in this work?\",\"answer\":\"k-Max DRVV selects at most k vertices to maximize the ratio between the number of dominated non-vulnerable vertices and the number of dominated vulnerable vertices, giving a combinatorial abstraction of vulnerability-aware influence trade-offs.\"},{\"question\":\"What approach and performance guarantees does the paper provide for solving these problems?\",\"answer\":\"The paper proposes an approximation algorithm based on an unbudgeted variant (DRVV), yielding an O(k/n)-approximation for bounded-degree graphs. For DSV it reduces to Red-Blue Set Cover and obtains a 2^|V|·(H(ΔN) − 1/2)-approximation, and for VCVE it gives a polynomial-time 2-approximation achieving the best possible ratio.\"}]",1784200383,60,{"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":86,"head_meta":88,"extra_data":90,"updated_unix":27},"domination-and-coverage-problems-under-vulnerability-constraints","",{"@graph":35,"@context":85},[36,53,68],{"@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/domination-and-coverage-problems-under-vulnerability-constraints/85022/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","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 is the motivation behind vulnerability constraints in domination and coverage problems?","Question",{"text":75,"@type":76},"The framework marks certain vertices or edges as vulnerable so they must not be dominated or covered. This models socially responsible objectives, and is also tied to influence maximization perspectives on network diffusion while controlling the impact on vulnerable elements.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How is the k-Max DRVV problem defined in this work?",{"text":80,"@type":76},"k-Max DRVV selects at most k vertices to maximize the ratio between the number of dominated non-vulnerable vertices and the number of dominated vulnerable vertices, giving a combinatorial abstraction of vulnerability-aware influence trade-offs.",{"name":82,"@type":73,"acceptedAnswer":83},"What approach and performance guarantees does the paper provide for solving these problems?",{"text":84,"@type":76},"The paper proposes an approximation algorithm based on an unbudgeted variant (DRVV), yielding an O(k/n)-approximation for bounded-degree graphs. 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