# Algorithms for weighted independent transversals and strong colouring

## TL;DR

This paper develops a widely applicable randomized algorithm for finding independent transversals in graphs, closing the gap between theoretical existence results and practical algorithms, and applies it to problems involving the strong chromatic number.

## Contribution

It introduces a new randomized algorithm for weighted independent transversals that extends previous deterministic methods, enabling efficient solutions for broader graph coloring problems.

## Key findings

- The randomized algorithm is more widely applicable than previous deterministic methods.
- Efficient algorithms are provided for problems related to the strong chromatic number.
- The approach narrows the gap between nonconstructive existence proofs and practical algorithms.

## Abstract

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.