[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86181-en":3,"doc-seo-86181-105":30,"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":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},86181,13056703019662,"Evangeline","https://ap-avatar.wpscdn.com/avatar/be000253a8e92610077?_k=1778726343310543188",8,"Research & Report","Bounded Support Additive Latin Transversals via Color Counted Matching","Bounded-support additive Latin transversals are studied over cyclic groups: given a multiset A in Zm and a set B of size k, elements of B are ordered so that all sums ai+bi become pairwise distinct. While full-size cases follow from Hall’s theorem and near-full cases remain constructible, general k lacks known polynomial-time constructions. A randomized algorithm is developed via a colored matching primitive, Color-Counted Matching, yielding randomized polynomial-time constructibility for every fixed support size.","arXiv :2607 . 1 124 1v 1 [ cs .DS] 13 Jul 2026  \nBounded-Support Additive Latin Transversals via Color-Counted Matching  \nAntoine Deza  \nMcMaster University, Hamilton, Ontario, Canada  \n[deza@mcmaster. ca](deza@mcmaster. ca)  \nYan Gerard  \nUniversité Clermont Auvergne, LIMOS, France  \n[yan. gerard@uca. fr](yan. gerard@uca. fr)  \nYijun Ma  \nMcMaster University, Hamilton, Ontario, Canada  \n[yijun@mcmaster. ca](yijun@mcmaster. ca)  \nSebastian Pokutta  \nZuse Institute Berlin and Technische Universität Berlin  \n[pokutta@zib. de](pokutta@zib. de)  \nJuly 14, 2026  \nAbstract  \nWe consider the following additive Latin transversal problem. Given a multiset A =(a1 ,..., ak) of elements of Zm and a set B ⊆ Zm of cardinality k, the task is to order B as b 1 ,..., bk so that the sums ai + bi are pairwise distinct. When k = m, Hall proved that a solution exists if and only if P ai ≡ 0 (mod m); moreover, his theorem yields a polynomial-time construction. Alon proved that a solution always exists when m is prime and k \u003C m, but no polynomial-time construction is known in general. Our main algorithmic contribution is a direct randomized algorithm for Color-Counted Matching: given an edge-colored graph and prescribed target counts for the colors, find a matching using exactly the prescribed number of edges of each color. If q is the sum of the target counts and h is the number of colors, our base-(q + 1) reduction to Exact Red Matching, combined with the algorithm of Mulmuley–Vazirani–Vazirani, gives a randomized algorithm with running time 􀀀|V |2 + |E| (q + 1)h−1􀀁 O(1) for an input graph (V, E) . Thus the dependence on the target matching size is qO (h), up to polynomial factors in the graph size. In contrast, applying the general matching-ILP theorem of Lassota and Ligthart as a black box yields a q O (h2 ) dependence for the corresponding fixed-size color-counted instances. Applying this primitive to additive Latin transversals with s = |supp(A)|, we obtain an algorithm in randomized time (k + log m)O (s) . In particular, additive Latin transversals are randomized polynomial-time constructible for every fixed support size.  \n1 Introduction  \nIn a January 2022 blog post, Eppstein highlighted the following open algorithmic question from a talk of Alon [4] . Fix a prime p, a sequence A = (a1 , ... , ak) ∈ (Zp)k with k \u003C p, and a k-subset B ⊆ Zp. Can one order the elements of B as b 1 ,..., bk so that the sums ai + bi are pairwise  \ndistinct? Alon proved that such an ordering always exists [1], but no polynomial-time construction is known. Consider the general case of an arbitrary cyclic group Zm. Over R, pairwise distinct sums can be obtained by ordering the elements of A and B increasingly. The full-size case where k = m is governed by Hall’s theorem [8]1 : For a sequence A = (a1 , ... , am) ∈ (Zm)m, there is an ordering b1 ,..., bm of all elements of Zm such that the sums ai+bi are pairwise distinct if and only if P ai ≡ 0 (mod m) . Hall’s theorem extends to the case when A and B have size k = m − 1. Hence the cases k = m and k = m − 1 are completely characterized with the important remark that the solutions are constructible in polynomial time [8] . Apart from the brute-force polynomiality for fixed k, the cases k = m and k = m − 1 are the only unrestricted regimes for which a polynomial-time construction is known. In this paper we study the algorithmic problem through the support size of the sequence A. Let supp(A) = {c0 ,..., cs−1} be the set of distinct values appearing in A, and lets = |supp(A)| . Our main result is that bounded support makes the problem tractable: for arbitrary modulus m, a solution can be found, whenever one exists, in randomized time (k + log m)O (s) . In particular, for every fixed support size s, additive Latin transversals are constructible in randomized polynomial time. The central algorithmic tool behind the result is a generic colored matching primitive that we call Color-Counted Matching. In this problem, the inp","cbCaij0lUlZXkIzl","https://ap.wps.com/l/cbCaij0lUlZXkIzl","pdf",444146,2,1,12,"English","en",105,"# Introduction\n# Related Results\n# Main Algorithmic Results\n## Reduction to Exact Red Matching\n## Bounded-Support Algorithm","[{\"question\":\"What is the additive Latin transversal ordering problem in the paper?\",\"answer\":\"The paper considers ordering the elements of B as b1,...,bk so that the sums ai+bi are pairwise distinct for a given multiset A in Zm.\"},{\"question\":\"What role does Hall’s theorem play in the full-size case?\",\"answer\":\"For k=m, Hall’s theorem characterizes existence: a valid ordering exists iff the sum of ai is congruent to 0 modulo m, and it also supports polynomial-time construction.\"},{\"question\":\"How does Color-Counted Matching help solve the bounded-support case?\",\"answer\":\"The paper reduces bounded-support additive Latin transversals to a colored matching problem where the algorithm finds a matching using exactly prescribed edge counts per color, enabling a randomized runtime bound of (k+log m)O(s) where s is the support 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is the additive Latin transversal ordering problem in the paper?","Question",{"text":75,"@type":76},"The paper considers ordering the elements of B as b1,...,bk so that the sums ai+bi are pairwise distinct for a given multiset A in Zm.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What role does Hall’s theorem play in the full-size case?",{"text":80,"@type":76},"For k=m, Hall’s theorem characterizes existence: a valid ordering exists iff the sum of ai is congruent to 0 modulo m, and it also supports polynomial-time construction.",{"name":82,"@type":73,"acceptedAnswer":83},"How does Color-Counted Matching help solve the bounded-support case?",{"text":84,"@type":76},"The paper reduces bounded-support additive Latin transversals to a colored matching problem where the algorithm finds a matching using exactly prescribed edge counts per color, enabling a randomized runtime bound of (k+log m)O(s) where s is the support 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