[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85160-en":3,"doc-seo-85160-105":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},85160,1374391974468,"Eden","https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0",8,"Research & Report","Complexity of partitioned-items response problems: matchings and perfect matchings","The paper studies bilevel optimization where a leader and follower jointly construct a feasible solution to an underlying combinatorial optimization problem. Response problems ask whether the leader can encourage or enforce a follower reaction that must include mandatory items and exclude forbidden items. Tractability is analyzed across combinations of mandatory, forbidden, and neutral items. After general results, matching and minimum-weight perfect matching are examined, revealing hardness even for a single mandatory or forbidden edge, plus fixed-parameter tractability for limited non-mandatory edges.","arXiv :2607 .09953v1 [math .OC] 10 Jul 2026  \nComplexity of partitioned-items response problems: matchings and perfect matchings  \nChristoph Buchheim1 , Lowig Duer1 , Eva Ley2 , Maximilian Merkert2 , and  \nKomal Muluk1  \n1 Department of Mathematics, TU Dortmund University, Germany  \n2 Institute for Mathematical Optimization, TU Braunschweig, Germany  \nAbstract  \nWe consider bilevel optimization problems in which leader and follower jointly construct a feasible solution for an underlying combinatorial optimization problem. Response problems ask whether the leader can encourage—or, in the pessimistic setting, enforce—a reaction of the follower that includes a set of mandatory items while excluding a set of forbidden items. Our investigation focuses on tractability results for various cases which emerge from different combinations of the total number of mandatory, forbidden, and neutral items. After providing some results for response problems that hold for any underlying combinatorial optimization problem, we examine response problems over the maximum-weight matching problem and the minimumweight perfect matching problem as illustrative and surprisingly varied examples. Among other results, we show that the response problem is hard for even a single given mandatory or forbidden edge. On the other hand, it is fixed-parameter tractable with respect to the total number of non-mandatory edges. If, however, each follower’s edge is either mandatory or forbidden, the response problem for the perfect matching problem is solvable in polynomial time while it remains NP-hard for the maximum-weight matching problem.  \n1 Introduction  \nIn bilevel optimization, two players solve a nested optimization problem in a hierarchical way: The leader takes a decision first, which influences the parameters of the follower’s optimization problem. The optimal solution of the latter in turn has an influence on the leader’s objective value and potentially also the feasibility of the leader’s decision. The leader thus has to anticipate the follower’s reaction when taking her decision, which often renders bilevel optimization problems significantly harder than their single-level counterparts. For instance, bilevel linear programs are known tobe NP-complete [14, 4] . For a general discussion of bilevel linear or mixed-integer optimization, including solution approaches, we refer to [8] or the very recent textbook [2] .  \nIn the context of combinatorial optimization, partitioned-items problems are a well-studied class of bilevel problems. Here, the ground set of the underlying problem is partitioned into items controlled by the leader and items controlled by the follower. First, the leader chooses a subset of her items. Then, the follower’s task is to extend it to a feasible solution of the underlying problem by means of appending a subset of his items. The leader aims to optimize her objective function while the follower aims to optimize his own objective function, which is in general different from that of the leader. For many underlying structures, this bilevel variant turns out to be one level harder in the polynomial hierarchy than their corresponding single-level counterparts. Among others, this has been shown for the assignment problem [11, 9], the spanning tree problem [19, 6], the knapsack problem [7], and the independent set problem [18] . For the partitioned-items shortest path problem, the bilevel variant is ΣP2-hard [12] . However, this is caused by the implicit fixing of edges in the shortest path problem to be solved by the follower or, equivalently, by the appearance of negative weights, in which case the shortest simple path problem itself is NP-hard [10] .  \nAnother branch of bilevel optimization problems asks whether the leader can incentivize a follower’s reaction that belongs to a given set of desired responses by modifying parameter values, e.g., objective coefficients. For linear programs, this decision problem has been investigated indi","cbCaiijnmhCZsy61","https://ap.wps.com/l/cbCaiijnmhCZsy61","pdf",341415,1,14,"English","en",105,"# Abstract\n# Introduction\n## Bilevel optimization and partitioned-items problems\n## Incentive, inverse, and partial inverse optimization\n## Response problems and complete vs partial responses\n# Complexity and tractability results","[{\"question\":\"What is a response problem in this paper’s bilevel optimization setting?\",\"answer\":\"The leader and follower jointly build a feasible solution of an underlying combinatorial problem, while the leader specifies mandatory items to be included and forbidden items to be excluded in the follower’s reaction.\"},{\"question\":\"How do the authors structure response problems using mandatory, forbidden, and neutral items?\",\"answer\":\"Items controlled by the follower are further classified by the leader as mandatory, forbidden, or neutral; with no neutral items the setting becomes a complete response problem, otherwise a partial response problem.\"},{\"question\":\"What complexity results are reported for matching and perfect matching response problems?\",\"answer\":\"The response problem is hard even with a single mandatory or forbidden edge, but it is fixed-parameter tractable with respect to the number of non-mandatory edges. For perfect matching, the response problem is solvable in polynomial time when every follower’s edge is mandatory or forbidden, while it remains NP-hard for maximum-weight matching.\"}]",1784201463,35,{"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},"complexity-of-partitioned-items-response-problems-matchings-and-perfect-matchings","",{"@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/complexity-of-partitioned-items-response-problems-matchings-and-perfect-matchings/85160/",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 a response problem in this paper’s bilevel optimization setting?","Question",{"text":75,"@type":76},"The leader and follower jointly build a feasible solution of an underlying combinatorial problem, while the leader specifies mandatory items to be included and forbidden items to be excluded in the follower’s reaction.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How do the authors structure response problems using mandatory, forbidden, and neutral items?",{"text":80,"@type":76},"Items controlled by the follower are further classified by the leader as mandatory, forbidden, or neutral; with no neutral items the setting becomes a complete response problem, otherwise a partial response problem.",{"name":82,"@type":73,"acceptedAnswer":83},"What complexity results are reported for matching and perfect matching response problems?",{"text":84,"@type":76},"The response problem is hard even with a single mandatory or forbidden edge, but it is fixed-parameter tractable with respect to the number of non-mandatory edges. For perfect matching, the response problem is solvable in polynomial time when every follower’s edge is mandatory or forbidden, while it remains NP-hard for maximum-weight matching.","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"]