[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81481-en":3,"doc-seo-81481-105":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},81481,1099513958607,"Jiven","https://ap-avatar.wpscdn.com/avatar/100002390cf8733938c?x-image-process=image/resize,m_fixed,w_180,h_180&k=1778829742770036399",8,"Research & Report","Finding Matchings in Dense Hypergraphs","Focusing on the algorithmic decision problem for n-vertex k-uniform hypergraphs, the work inputs a minimum codegree promise of at least m−c and determines whether a matching of size m exists. A fixed-parameter tractable algorithm with parameter c is presented. The procedure goes beyond yes/no: it outputs either a matching of size m or a certificate proving none exists. When m=n/k and c=O(log n), the result yields a polynomial-time algorithm that finds a perfect matching or certifies its absence.","arXiv :2210 . 12643v2 [math .CO] 10 Jul 2026  \nFINDING MATCHINGS IN DENSE HYPERGRAPHS  \nJIE HAN AND PETER KEEVASH  \nAbstract. We consider the algorithmic decision problem that takes as input an n-vertex k-uniform hypergraph H with minimum codegree at least m − c and decides whether it has a matching of size  \nm. We show that this decision problem is fixed parameter tractable with respect to c. Furthermore, our algorithm not only decides the problem, but actually either finds a matching of size m or a certificate that no such matching exists. In particular, when m = n/k and c = O(log n), this gives a polynomial-time algorithm, that given any n-vertex k-uniform hypergraph H with minimum codegree at least n/k−c, finds either a perfect matching in H or a certificate that no perfect matching exists.  \n1. Introduction  \nMatchings are fundamental objects in Graph Theory and have broad applications in other branches of Science and a variety of practical problems (e.g. the assignment of graduating medical students to their first hospital appointments 1) . Applications of matchings in hypergraphs include the ‘Santa Claus’ allocation problem [3]; they also offer a universal framework for many important combinatorial problems, e.g. the Existence Conjecture for designs (see [10, 20]) and Ryser’s conjecture [34] on transversals in Latin squares.  \nThis paper is concerned with the algorithmic question of finding a matching that is perfect, meaning that it covers all vertices of the graph or hypergraph. The graph case of this question is well understood: Tutte’s Theorem [39] gives necessary and sufficient conditions for a graph to contain a perfect matching, and Edmonds’ Algorithm [5] finds such a matching in polynomial time. However, for hypergraphs it is a different story: in fact, determining whether a 3-uniform hypergraph contains a perfect matching was one of Karp’s celebrated 21 NP-complete problems [17] . As the general problem is intractable (assuming P ≠ NP), it is natural to seek conditions that guarantee a perfect matching, or at least make the existence question tractable.  \n1.1. Perfect matchings under minimum degree conditions. We start with some definitions that will be used throughout the paper. Given k ≥ 2, a k-uniform hypergraph (in short, k-graph) H = (V, E) consists of a vertex set V and an edge set E ⊆ (Vk) , where every edge is a k-element subset of V. A matching in H is a collection of vertex-disjoint edges of H. A perfect matching Min H is a matching that covers all vertices of H. We always assume that k divides n ∶= ∣V(H)∣ , which is clearly a necessary condition for the existence of a perfect matching in H. The perfect matching problem is also known as the exact cover problem and can be viewed as a stricter form of the set cover problem. Finding a matching of certain size is known as the k-set packing problem: the decision problem for the existence of a given number of disjoint subsets from a family F of sets of size at most k. See also the survey [16] .  \nWe will consider the algorithmic effect of minimum degree conditions defined as follows. For S ⊆ V(H) the neighbourhood of S is NH(S) ∶= {T ⊆ V(H) ∖ S ∶ S ∪ T ∈ E(H)}, and the degree of  \nSupported by ERC Advanced Grant 883810 .  \n1In 2012, the Nobel Memorial Prize in Economics was awarded to Shapley and Roth “for the theory of stable allocationsand the practice of market design.”  \nS is degH (S) = ∣NH(S)∣; the subscript H is omitted if it is clear from the context. The minimum d-degree δd(H) of H is the minimum of degH (S) over all d-vertex sets S in H.  \nWe refer to δk−1(H) as the minimum codegree of H. This is the usual parameter considered for minimum degree results on hypergraph matchings. It is also natural to consider other d, although the case d = 0, which concerns the size of H, leads to a rather trivial question, as it is not hard to see that the largest size is attained by a complete hypergraph on n − 1 vertices together with an isolated vertex.  \nR¨odl,","cbCaitAhbNPEwoR5","https://ap.wps.com/l/cbCaitAhbNPEwoR5","pdf",485937,1,20,"English","en",105,"# Introduction\n## Perfect matchings under minimum degree conditions\n## Algorithms","[{\"question\":\"What is the main algorithmic problem studied for dense hypergraphs?\",\"answer\":\"Given an n-vertex k-uniform hypergraph with minimum codegree at least m−c, the paper studies the decision problem of whether it contains a matching of size m, with an emphasis on perfect matchings.\"},{\"question\":\"What does the proposed algorithm guarantee besides the yes/no decision?\",\"answer\":\"The algorithm is fixed parameter tractable in c and outputs either a matching of size m or a certificate that no such matching exists.\"},{\"question\":\"Under what parameter regime does the paper obtain a polynomial-time algorithm for perfect matchings?\",\"answer\":\"When m=n/k and c=O(log n), it yields a polynomial-time algorithm that either finds a perfect matching or certifies that no perfect matching exists, assuming minimum codegree at least 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is the main algorithmic problem studied for dense hypergraphs?","Question",{"text":75,"@type":76},"Given an n-vertex k-uniform hypergraph with minimum codegree at least m−c, the paper studies the decision problem of whether it contains a matching of size m, with an emphasis on perfect matchings.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What does the proposed algorithm guarantee besides the yes/no decision?",{"text":80,"@type":76},"The algorithm is fixed parameter tractable in c and outputs either a matching of size m or a certificate that no such matching exists.",{"name":82,"@type":73,"acceptedAnswer":83},"Under what parameter regime does the paper obtain a polynomial-time algorithm for perfect matchings?",{"text":84,"@type":76},"When m=n/k and c=O(log n), it yields a polynomial-time algorithm that either finds a perfect matching or certifies that no perfect matching exists, assuming minimum codegree at least 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