A Linear Time Parameterized Algorithm for Node Unique Label Cover

TL;DR

This paper introduces the first linear-time fixed-parameter tractable algorithm for the Node Unique Label Cover problem, advancing the understanding of its computational complexity and providing efficient solutions.

Contribution

It presents the first linear-time FPT algorithm for Node Unique Label Cover, solving an open problem in parameterized complexity.

Findings

The algorithm runs in linear time with respect to input size.

It improves the parameter dependence over previous algorithms.

It advances the theoretical understanding of the problem's complexity.

Abstract

The optimization version of the Unique Label Cover problem is at the heart of the Unique Games Conjecture which has played an important role in the proof of several tight inapproximability results. In recent years, this problem has been also studied extensively from the point of view of parameterized complexity. Cygan et al. [FOCS 2012] proved that this problem is fixed-parameter tractable (FPT) and Wahlstr\"om [SODA 2014] gave an FPT algorithm with an improved parameter dependence. Subsequently, Iwata, Wahlstr\"om and Yoshida [2014] proved that the edge version of Unique Label Cover can be solved in linear FPT-time. That is, there is an FPT algorithm whose dependence on the input-size is linear. However, such an algorithm for the node version of the problem was left as an open problem. In this paper, we resolve this question by presenting the first linear-time FPT algorithm for Node Unique Label Cover.