[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85364-en":3,"doc-seo-85364-105":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},85364,7971461741311,"Ophelia","https://ap-avatar.wpscdn.com/avatar/74000253aff267980c6?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779345379180704826",8,"Research & Report","Representing the Non-dominated Set of Multi-objective Network Problems by Supported Non-dominated Points","Multi-objective combinatorial optimization produces a non-dominated set that is often dominated in size and computational difficulty by unsupported points. Prior work indicates that extreme supported non-dominated points can represent the non-dominated set well for some binary problems. This generalization fails for capacitated network optimization: representation quality degrades with arc capacities, while supported points remain consistently strong across multiple quality indicators. Fixed-size subset representations can be expensive, so the paper proposes supported points as candidate sets for subset selection, yielding nearly equal quality.","arXiv :2607 . 11821v1 [ cs .DM] 13 Jul 2026  \nRepresenting the Non-dominated Set of Multi-objective Network Problems by Supported Non-dominated Points  \nDavid Könen 1,2 , Lara Löhken 1,3 , Michael Stiglmayr 1,4  \n1 University of Wuppertal, School of Mathematics and Natural Sciences, Optimization Group, Gaußstraße 20, 42103 Wuppertal, Germany,  \n2 Corresponding author, E-Mail: [koenen@uni-wuppertal.de](koenen@uni-wuppertal.de), ORCID-ID: 0000-0003-1747-8791  \n3 E-Mail: [loehken@uni-wuppertal.de](loehken@uni-wuppertal.de), ORCID-ID: 0009-0003-9205-6641  \n4 E-Mail: [stiglmayr@uni-wuppertal.de](stiglmayr@uni-wuppertal.de), ORCID-ID: 0000-0003-0926-1584  \nIn multi-objective combinatorial optimization, unsupported non-dominated points typically outnumber supported points and are often significantly more challenging to compute. Recent studies show that extreme supported non-dominated points provide high-quality representations of the non-dominated set for certain binary problems. We demonstrate that this observation does not generalize to capacitated network optimization problems: representation quality decreases with increasing arc capacities, whereas supported non-dominated points consistently provide highquality representations with respect to several quality indicators. However, supported point sets may still be too large in practical applications, where only a small, fixed number of alternatives is typically desired. Selecting fixed-size representations from the non-dominated set requires its computationally expensive generation and thus diminishes the computational advantages that representations are intended to provide. We therefore suggest the (extreme) supported points as alternative candidate sets in subset selection problems. Our numerical results show that restricting the candidate set to supported non-dominated points yields fixed-size representations of nearly the same quality as those selected from the complete non-dominated set. Overall, supported non-dominated points serve both as high-quality representations and as reasonable candidate sets for subset selection.  \nKeywords: multi-objective optimization | combinatorial optimization | representation | supportedness | network optimization | minimum-cost flow problem  \n1 Introduction  \nNetwork optimization problems are fundamental optimization problems, arising when a commodity moves through an underlying network, creating a flow of resources or in-  \nformation. Nowadays, such networks appear in various forms in our daily lives. The textbook by Ahuja et al. (1993) presents over 150 applications of network optimization problems across various fields, such as engineering, management, and other scientific domains. While single-objective variants have been studied since the early 1950s (see Ahuja et al., 1993; Bertsekas, 1998, and the references given therein), real-world applications, however, often involve conflicting objectives.  \nThese problems belong to the general class of multi-objective integer linear problems (MOILP), defined as  \nmin z(x) = (z1 (x), . . . , zp(x))⊤ = Cx, (MOILP) x∈X  \nwith the cost matrix C ∈ Zp×n containing the rows ck of coefficients of p ≥ 2 linear objective functions z k(x) = ck x for k ∈ {1, . . . , p} and the feasible set X := {x ∈ Zn : Ax = b}, where A ∈ Zm×n , b ∈ Zm describe the m constraints. If x ∈ {0, 1}n these problems are also referred to multi-objective combinatorial optimization (MOCO) problems. The set of feasible outcome vectors in the objective space is denoted by Y := {C x : x ∈ X }, where X is the set of all feasible solutions.  \nIn multi-objective optimization, it is usually assumed that the objective functions are conflicting, which implies that no solution minimizes all objectives simultaneously. In this situation, we use the Pareto concept of optimality, based on the component-wise orders in Rp. Let y 1 , y 2 ∈ Rp then  \ny 1 ≦ y 2 ⇐⇒ y1k ≤ y2k ∀ k ∈ {1, . . . , p} y 1 ⩽ y 2 ⇐⇒ y 1 ≦ y 2 and y 1  y 2 y 1 \u003C y 2 ⇐⇒ y1k \u003C y2k ","cbCaijbocKZRUtBd","https://ap.wps.com/l/cbCaijbocKZRUtBd","pdf",690381,1,33,"English","en",105,"# Introduction\n# Multi-objective optimization background\n# Scalarization and Supported Solutions","[{\"question\":\"Why are unsupported non-dominated points difficult to compute in multi-objective combinatorial optimization?\",\"answer\":\"Unsupported non-dominated points typically outnumber supported points and are often significantly more challenging to compute, making representation of the non-dominated set harder in practice.\"},{\"question\":\"Does the previously observed effectiveness of extreme supported non-dominated points generalize to capacitated network optimization problems?\",\"answer\":\"No. For capacitated network optimization problems, representation quality decreases as arc capacities increase, so the earlier favorable observation does not carry over.\"},{\"question\":\"How do supported non-dominated points help when fixed-size representations are required?\",\"answer\":\"Although supported candidate sets may still be large, generating the full non-dominated set to pick a fixed number of alternatives is computationally expensive. Using supported non-dominated points as the candidate set enables fixed-size representations that achieve nearly the same quality as those selected from the complete non-dominated set.\"}]",1784202797,83,{"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},"representing-the-non-dominated-set-of-multi-objective-network-problems-by-supported-non-dominated-points","",{"@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/representing-the-non-dominated-set-of-multi-objective-network-problems-by-supported-non-dominated-points/85364/",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},"Why are unsupported non-dominated points difficult to compute in multi-objective combinatorial optimization?","Question",{"text":75,"@type":76},"Unsupported non-dominated points typically outnumber supported points and are often significantly more challenging to compute, making representation of the non-dominated set harder in practice.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Does the previously observed effectiveness of extreme supported non-dominated points generalize to capacitated network optimization problems?",{"text":80,"@type":76},"No. For capacitated network optimization problems, representation quality decreases as arc capacities increase, so the earlier favorable observation does not carry over.",{"name":82,"@type":73,"acceptedAnswer":83},"How do supported non-dominated points help when fixed-size representations are required?",{"text":84,"@type":76},"Although supported candidate sets may still be large, generating the full non-dominated set to pick a fixed number of alternatives is computationally expensive. Using supported non-dominated points as the candidate set enables fixed-size representations that achieve nearly the same quality as those selected from the complete non-dominated set.","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":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":45,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":45,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":45,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":45,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":45,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]