AI 中文总结
研究加法拉丁横截问题，核心方法是给出颜色计数匹配的随机算法，通过归约和结合已有算法得到运行时间，主要贡献是对加法拉丁横截问题，在固定支持大小时可随机多项式时间构造。
AI 中文摘要
我们考虑以下加法拉丁横截问题。给定\(\mathbb{Z}_m\)中元素的多重集\(A=(a_1,\dots,a_k)\)和基数为\(k\)的集合\(B\subseteq\mathbb{Z}_m\)，任务是将\(B\)排序为\(b_1,\dots,b_k\)，使得和\(a_i + b_i\)两两不同。当\(k = m\)时，Hall证明当且仅当\(\sum_{i = 1}^m a_i\equiv 0\pmod{m}\)时存在解，且其定理给出了多项式时间构造。Alon证明当\(m\)为素数且\(k < m\)时总是存在解，但一般没有多项式时间构造。我们主要的算法贡献是一种用于颜色计数匹配的直接随机算法：给定一个边着色图和规定的颜色目标计数，找到一个使用每种颜色恰好规定数量边的匹配。若\(q\)是目标计数之和且\(h\)是颜色数，我们基于\((q + 1)\)归约到精确红色匹配，并结合Mulmuley-Vazirani-Vazirani算法，给出一个运行时间为\(\left(|V|^2+|E|(q + 1)^{h - 1}\right)^{O(1)}\)的随机算法。将此原语应用于加法拉丁横截，当\(s = |\text{supp}(A)|\)时，我们得到一个随机时间为\((k+\log m)^{O(s)}\)的算法。特别地，对于每个固定的支持大小，加法拉丁横截是随机多项式时间可构造的。
英文摘要
We consider the following additive Latin transversal problem. Given a multiset $A=(a_1,\dots,a_k)$ of elements of $\mathbb Z_m$ and a set $B\subseteq\mathbb Z_m$ of cardinality $k$, the task is to order $B$ as $b_1,\dots,b_k$ so that the sums $a_i+b_i$ are pairwise distinct. When $k=m$, Hall proved that a solution exists if and only if $\sum_{i=1}^m a_i\equiv 0 \pmod m$; moreover, his theorem yields a polynomial-time construction. Alon proved that a solution always exists when $m$ is prime and $k<m$, 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 $\left(|V|^2+|E|(q+1)^{h-1}\right)^{O(1)} $ for an input graph $(V,E)$. Thus the dependence on the target matching size is $q^{O(h)}$, 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 $q^{O(h^2)}$ dependence for the corresponding fixed-size color-counted instances. Applying this primitive to additive Latin transversals with $s=|\operatorname{supp}(A)|$, 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.
