An independent transversal (IT) in a graph with a given vertex partition is an independent set consisting of one vertex in each partition class. Several sufficient conditions are known for the existence of an IT in a given graph with a given vertex partition, which have been used over the years to solve many combinatorial problems. Some of these IT existence theorems have algorithmic proofs, but there remains a gap between the best bounds given by nonconstructive results, and those obtainable by efficient algorithms. Recently, Graf and Haxell (2018) described a new (deterministic) algorithm that asymptotically closes this gap, but there are limitations on its applicability. In this paper we develop a randomized version of this algorithm that is much more widely applicable, and demonstrate its use by giving efficient algorithms for two problems concerning the strong chromatic number of graphs.

中文翻译：

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加权独立横向和强着色的算法

具有给定顶点分区的图中的独立横向 (IT) 是由每个分区类中的一个顶点组成的独立集合。对于在给定顶点分区的给定图中存在 IT 的几个充分条件是已知的，这些条件多年来已被用于解决许多组合问题。其中一些 IT 存在定理具有算法证明，但非构造性结果给出的最佳界限与有效算法获得的最佳界限之间仍然存在差距。最近，Graf 和 Haxell (2018) 描述了一种新的（确定性）算法，该算法渐近地缩小了这一差距，但其适用性存在局限性。在本文中，我们开发了该算法的随机版本，该版本适用范围更广，