A Note on Finding Large Transversals Efficiently

IF 0.5 4区数学 Q3 MATHEMATICS

Journal of Combinatorial Designs Pub Date : 2025-05-26 DOI:10.1002/jcd.21990

Michael Anastos, Patrick Morris

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引用次数: 0

Abstract

In an $n \times n$ array filled with symbols, a transversal is a collection of entries with distinct rows, columns and symbols. In this note we show that if no symbol appears more than $β n$ times, the array contains a transversal of size $(1 - β ∕ 4 - o (1)) n$ . In particular, if the array is filled with $n$ symbols, each appearing $n$ times (an equi- $n$ square), we get transversals of size $(3 ∕ 4 - o (1)) n$ . Moreover, our proof gives a deterministic algorithm with polynomial running time, that finds these transversals.

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关于高效查找大截线的注释

在一个充满符号的n × n数组中，截线是具有不同行、列和符号的条目的集合。在这个笔记中，我们表明如果没有符号出现超过β n次，该阵列包含大小为（1−β∕4）的横截面−0 (1))n。特别是，如果数组被n个符号填充，每个符号出现n次（一个相等- n的平方），我们得到大小为（3∕4−0）的截线(1)；此外，我们的证明给出了一个运行时间为多项式的确定性算法，可以找到这些截线。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Combinatorial Designs 数学-数学

CiteScore

1.60

自引率

14.30%

发文量

审稿时长

>12 weeks

期刊介绍： The Journal of Combinatorial Designs is an international journal devoted to the timely publication of the most influential papers in the area of combinatorial design theory. All topics in design theory, and in which design theory has important applications, are covered, including: block designs, t-designs, pairwise balanced designs and group divisible designs Latin squares, quasigroups, and related algebras computational methods in design theory construction methods applications in computer science, experimental design theory, and coding theory graph decompositions, factorizations, and design-theoretic techniques in graph theory and extremal combinatorics finite geometry and its relation with design theory. algebraic aspects of design theory. Researchers and scientists can depend on the Journal of Combinatorial Designs for the most recent developments in this rapidly growing field, and to provide a forum for both theoretical research and applications. All papers appearing in the Journal of Combinatorial Designs are carefully peer refereed.