[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84072-en":3,"doc-seo-84072-105":29,"detail-sidebar-cat-0-en-105":90},{"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":21,"is_downloadable":21,"audit_status":21,"page_count":11,"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},84072,1099514067415,"Rowan","https://ap-avatar.wpscdn.com/avatar/100002539d78ffe74a7?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779092875211072502",8,"Research & Report","The Singleton Hypergraph Is Extremal for the Isolation Lemma","Let H be an inclusion-free hypergraph on n vertices. A weight assignment w:[n]→[d] is isolating if exactly one edge e achieves the minimum total edge weight, i.e., w(e)=∑i∈e w(i) is minimal among all edges. The work proves a tight lower bound on the number of isolating assignments, attaining equality for the hypergraph formed by the n singleton edges. It also extends the result to broader edge-weight objectives including arbitrary offsets.","The singleton hypergraph is extremal for the Isolation Lemma  \nVance Faber∗, David G. Harris†  \narXiv :2607 .06171v1 [math .CO] 7 Jul 2026  \nAbstract  \nLet H be an inclusion-free hypergraph on n vertices. A weight assignment w : [n] → [d] is isolating if there is a unique edge e whose weight w(e) = Pi∈e w (i) is minimum. We show that the number of isolating weight assignments is at least  \nd−1  \nnX jn−1 ,  \nj=0  \na bound which is attained with equality by the hypergraph consisting of the n singleton edges.  \nThis proves the conjecture stated in Faber & Harris (2018) .  \nWe also prove the bound for a more general class of edge-weight objectives, including arbitrary edge offsets.  \n1 Introduction  \nThe Isolation Lemma, introduced by Mulmuley, Vazirani, and Vazirani [2], is a basic tool in randomized algorithms. In one common form, one has a hypergraph H whose edges encode the possible solutions to a search problem. A random weight assignment to the vertices is chosen, and one hopes there is a unique edge of minimum weight. Such a weight assignment is called isolating. When this occurs, the original search problem has been reduced to a unique-solution problem.  \nThe classical Isolation Lemma in [2] gives a lower bound of dn (1 − n/d) for the number of isolating weight assignments. Ta-Shma [4] improved the bound to (d − 1)n via a simple and elegant argument. This was investigated further by Faber & Harris [1], with a number of improved bounds; in particular, they conjectured that the minimum is attained by the singleton hypergraph, whose edges are {1},..., {n} . They showed the conjecture for several special cases as well as an asymptotic regime.  \nIn this note, we fully prove the conjecture of [1], giving the tight extremal bound on the number of isolating assignments. We will work in a slightly more general setting. Throughout, H is anon-empty inclusion-free hypergraph on vertex set [n] = {1,..., n} . For a function f : H → R and weight assignment w : [n] → Z, we define the weight of an edge e by  \nfw (e) := f(e) +X w(i) i∈e  \nFor d ≥ 1 write [d] = {1,..., d} . The main theorem is the following.  \n∗ Hoquiam, WA. Email: [vance.faber@gmail.com](vance.faber@gmail.com)  \n†University of Maryland, Dept. of Computer Science. Email: [davidgharris29@gmail.com](davidgharris29@gmail.com)  \nTheorem 1 . Let d, n ≥ 1. For every H, f as above, the number of isolating weight assignments w ∈ [d]n on H is at least  \nd−1  \nnX jn−1 .  \nj=0  \nThis bound is tight for every d, n, attained by the singleton hypergraph with f (e) = 0 for all e.(Here and throughout we use the convention 00 = 1 to cover a few edge cases in the formulas.)  \n2 Definitions and basic reductions  \nWe define MH,f(w) to be the set of minimum-weight edges, i.e.  \nMH,f(w) = {e ∈ H : fw (e) ≤ fw (h) for every h ∈ H} .  \nThe weight assignment w is isolating if |MH,f(w)| = 1, that is, there is exactly one minimum-weight edge. In that case, the unique minimum-weight edge is called the isolated edge. Note that, since His inclusion-free, the set family MH,f(w) is necessarily an antichain.  \nFor a set S ⊆ [n], we define the indicator vector for S by χS , i.e.  \nχS (i) = (10  \nfor i ∈ S  \nfor i  S  \nThe key insight of [4] was that an arbitrary weight assignment could be transformed into an isolating assignment by subtracting the indicator of a minimum-weight edge.  \nProposition 2 ([4]) . Let e ∈ MH,f(w) for a weight assignment w. Then the weight assignment  \nu = w − χe  \nis isolating, with isolated edge e.  \nProof. Let h ∈ H with h  e. Then  \nfu (h) − fu(e) = fw(h) − fw(e) + |e \\ h|  \nIf h  MH,f(w), then fw (h) − fw(e) > 0 strictly. If h ∈ MH,f(w), then e \\ h  ∅ since H is inclusion-free. In all cases, fu (h) > fu (e), and e is the unique minimum edge under u.   \nOur proof will need an extension of Proposition 2, using the concept of private pairs in an antichain.  \nDefinition 3 . Let A be an antichain. We call (A, S) a private pair of A if A ∈ A, S ⊆ A, and  \nS ⊈ B for every B ∈ A \\ {A}  ","cbCaikU3nX5z2RXX","https://ap.wps.com/l/cbCaikU3nX5z2RXX","pdf",311096,3,1,"English","en",105,"# Introduction\n## Isolation Lemma background\n## Prior bounds and conjecture\n# Definitions and basic reductions\n## Minimum-weight edges and isolating assignments\n## Reductions via indicator subtraction\n# Layering approach\n## Layer-one bounds\n## Proof strategy for the main theorem","[{\"question\":\"What does it mean for a weight assignment to be isolating in this paper?\",\"answer\":\"A weight assignment w is isolating if there is exactly one edge e whose total weight w(e)=∑i∈e w(i) is minimal among all edges of the inclusion-free hypergraph H.\"},{\"question\":\"What hypergraph is shown to attain the extremal (tight) bound?\",\"answer\":\"The extremal bound is attained with equality by the hypergraph consisting of the n singleton edges {1},...,{n}, with the weight function f≡0.\"},{\"question\":\"How does the paper extend beyond the classical Isolation Lemma setting?\",\"answer\":\"It proves the bound for a more general class of edge-weight objectives, including arbitrary edge offsets, using generalized reduction 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does it mean for a weight assignment to be isolating in this paper?","Question",{"text":74,"@type":75},"A weight assignment w is isolating if there is exactly one edge e whose total weight w(e)=∑i∈e w(i) is minimal among all edges of the inclusion-free hypergraph H.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"What hypergraph is shown to attain the extremal (tight) bound?",{"text":79,"@type":75},"The extremal bound is attained with equality by the hypergraph consisting of the n singleton edges {1},...,{n}, with the weight function f≡0.",{"name":81,"@type":72,"acceptedAnswer":82},"How does the paper extend beyond the classical Isolation Lemma setting?",{"text":83,"@type":75},"It proves the bound for a more general class of edge-weight objectives, including arbitrary edge offsets, using generalized reduction 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