[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84114-en":3,"doc-seo-84114-105":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},84114,1099514067438,"River Wang","https://ap-avatar.wpscdn.com/avatar/100002539ee87300030?x-image-process=image/resize,m_fixed,w_180,h_180&k=1780474512215547542",8,"Research & Report","FO Value Discovery and Partial Vertex Cover Discovery","Study solution discovery in the token-sliding model using logical and cost-value optimization. Given a graph, an initial placement of k tokens, and a movement budget b, the goal is to reach a target configuration satisfying a prescribed condition. Building on Partial Vertex Cover Discovery, the work introduces FO Value Discovery where tuple value combines unary vertex weights with first-order definable correction terms. It further defines FO Cost-Value Decision and proves a conditional metatheorem via local cost-value decision and anchored multicolored distance-independence, yielding complexity results for restricted graph classes and logic fragments.","arXiv :2607 .06446v 1 [ cs .DM] 7 Jul 2026  \nFO Value Discovery and Partial Vertex Cover Discovery  \nEnna Gerhard   \nUniversity of Bremen, Germany  \nStephanie Maaz   \nUniversity of Waterloo, Canada  \nPascale Schott   \nUniversity of Bremen, Germany  \nSebastian Siebertz   \nUniversity of Bremen, Germany  \nJan Wodtke   \nUniversity of Bremen, Germany  \n~~ Abstract ~~  \nWe study solution discovery in the token-sliding model from a logical and cost-value optimization perspective. In solution discovery, we are given a graph, an initial placement of k tokens, and a movement budget b. The task is to find a reachable target configuration satisfying a prescribed condition.  \nOur results are inspired by Partial Vertex Cover Discovery, where the condition is that the k tokens cover at least t edges of the input graph. This objective is not merely a sum of independent occupied vertex contributions: each selected vertex contributes its degree, but edges with both endpoints selected have to be subtracted once. To capture this phenomenon, we introduce FO Value Discovery, an optimization problem in which the value of a selected tuple is given byunary vertex weights together with first-order definable correction terms. This framework generalizes Boolean FO Discovery and contains Partial Vertex Cover Discovery as a quantifier-free special case.  \nWe further generalize the setting to FO Cost-Value Decision, where vertices carry both costs and values, and the task is to decide whether there is a tuple whose first-order value expression reaches a prescribed value threshold while respecting a cost bound. We prove a conditional metatheorem based on two algorithmic ingredients: local FO cost-value decision and an anchored weighted multicolored distance-independence problem. We then prove that these assumptions are satisfied in several restricted graph classes and fragments of first-order logic.  \nFinally, we study the parameterized complexity of Partial Vertex Cover Discovery and Vertex Cover Discovery. As a consequence of the logical meta-theorems, we obtain fixedparameter tractability of Partial Vertex Cover Discovery on several graph classes, including classes of locally bounded cliquewidth. We also show that Partial Vertex Cover Discovery is W[1]-hard parameterized by k+b and fixed-parameter tractable on d-degenerate graphs parameterized by k + d. For Vertex Cover Discovery, we prove NP-hardness on planar graphs, W[1]-hardness parameterized by the clique cover number, even when a clique cover is supplied with the input, and W[1]-hardness with respect to parameter cutwidth.  \n2012 ACM Subject Classification Theory of computation → Parameterized complexity and exact algorithms; Mathematics of computing → Graph theory  \nKeywords and phrases solution discovery, first-order logic, meta theorems, partial vertex cover  \n 1  Introduction  \nMany graph problems ask for a set (or tuple) of vertices with a prescribed property, such as being a vertex cover or an independent set. In solution discovery, one does not construct  \n2 FO Value and Partial Vertex Cover Discovery  \nsuch a solution from scratch. Instead, the input already contains an initial placement of tokens, and the task is to decide whether the tokens can be moved, within a given budget, toa placement that satisfies the desired property. This captures situations in which a current undesired assignment or configuration is established, and only a limited number of local repair operations are allowed to reach a feasible solution.  \nThe solution discovery viewpoint was formalized by Fellows et al. [10] and is closely related to earlier work on minimizing movement to reach a desired configuration, including the work of Demaine et al. [6] . It has since been studied for polynomial-time solvable graph problems [14], from the viewpoint of kernelization [15], through algorithmic meta-theorems [1], for several classical vertex-subset problems [31], and in a two-graph model that separates feasibility from movem","cbCaicvZgD4t5TSm","https://ap.wps.com/l/cbCaicvZgD4t5TSm","pdf",693910,1,29,"English","en",105,"# Introduction\n# FO Value and Partial Vertex Cover Discovery","[{\"question\":\"What problem is studied in the token-sliding model?\",\"answer\":\"The paper studies solution discovery, where an initial placement of k tokens on a graph must be moved within a budget b to a reachable configuration that satisfies a prescribed condition.\"},{\"question\":\"How does FO Value Discovery extend Partial Vertex Cover Discovery?\",\"answer\":\"Partial Vertex Cover uses coverage of at least t edges by selected vertices, with a subtraction for edges whose both endpoints are selected. FO Value Discovery generalizes this by defining tuple value using unary vertex weights plus first-order definable correction terms.\"},{\"question\":\"What complexity results are proved for Partial Vertex Cover Discovery and Vertex Cover Discovery?\",\"answer\":\"The paper shows fixed-parameter tractability of Partial Vertex Cover Discovery on several graph classes, including locally bounded cliquewidth and d-degenerate graphs with parameter k+d, and W[1]-hardness parameterized by k+b. For Vertex Cover Discovery, it proves NP-hardness on planar graphs and additional W[1]-hardness results under other parameters such as clique cover number and cutwidth.\"}]",1784192942,73,{"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},"fo-value-discovery-and-partial-vertex-cover-discovery","",{"@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/fo-value-discovery-and-partial-vertex-cover-discovery/84114/",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 problem is studied in the token-sliding model?","Question",{"text":75,"@type":76},"The paper studies solution discovery, where an initial placement of k tokens on a graph must be moved within a budget b to a reachable configuration that satisfies a prescribed condition.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does FO Value Discovery extend Partial Vertex Cover Discovery?",{"text":80,"@type":76},"Partial Vertex Cover uses coverage of at least t edges by selected vertices, with a subtraction for edges whose both endpoints are selected. FO Value Discovery generalizes this by defining tuple value using unary vertex weights plus first-order definable correction terms.",{"name":82,"@type":73,"acceptedAnswer":83},"What complexity results are proved for Partial Vertex Cover Discovery and Vertex Cover Discovery?",{"text":84,"@type":76},"The paper shows fixed-parameter tractability of Partial Vertex Cover Discovery on several graph classes, including locally bounded cliquewidth and d-degenerate graphs with parameter k+d, and W[1]-hardness parameterized by k+b. For Vertex Cover Discovery, it proves NP-hardness on planar graphs and additional W[1]-hardness results under other parameters such as clique cover number and cutwidth.","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"]