# Lower Bounds for the Happy Coloring Problems

## TL;DR

This paper investigates the computational complexity of the Maximum Happy Vertices and Edges problems, establishing lower bounds and presenting algorithms that match these bounds, thus clarifying their computational limits.

## Contribution

It provides new lower bounds on the complexity of the problems and introduces algorithms with proven optimality under current complexity assumptions.

## Key findings

- NP-hardness of guarantee parameterization
- Kernelization lower bounds
- Exponential lower bounds under Set Cover and ETH

## Abstract

In this paper, we study the Maximum Happy Vertices and the Maximum Happy Edges problems (MHV and MHE for short). Very recently, the problems attracted a lot of attention and were studied in Agrawal '17, Aravind et al. '16, Choudhari and Reddy '18, Misra and Reddy '17. Main focus of our work is lower bounds on the computational complexity of these problems. Established lower bounds can be divided into the following groups: NP-hardness of the above guarantee parameterization, kernelization lower bounds (answering questions of Misra and Reddy '17), exponential lower bounds under the Set Cover Conjecture and the Exponential Time Hypothesis, and inapproximability results. Moreover, we present an $\mathcal{O}^*(\ell^k)$ randomized algorithm for MHV and an $\mathcal{O}^*(2^k)$ algorithm for MHE, where $\ell$ is the number of colors used and $k$ is the number of required happy vertices or edges. These algorithms cannot be improved to subexponential taking proved lower bounds into account.