Approximation algorithms for the vertex happiness

We investigate the maximum happy vertices (MHV) problem and its complement, the minimum unhappy vertices (MUHV) problem. We first show that the MHV and MUHV problems are a special case of the supermodular and submodular multi-labeling (Sup-ML and Sub-ML) problems, respectively, by re-writing the objective functions as set functions. The convex relaxation on the Lov\'{a}sz extension, originally presented for the submodular multi-partitioning (Sub-MP) problem, can be extended for the Sub-ML problem, thereby proving that the Sub-ML (Sup-ML, respectively) can be approximated within a factor of $2 - \frac{2}{k}$ ($\frac{2}{k}$, respectively). These general results imply that the MHV and the MUHV problems can also be approximated within $\frac{2}{k}$ and $2 - \frac{2}{k}$, respectively, using the same approximation algorithms. For MHV, this $\frac{2}{k}$-approximation algorithm improves the previous best approximation ratio $\max \{\frac{1}{k}, \frac{1}{\Delta + 1}\}$, where $\Delta$ is the maximum vertex degree of the input graph. We also show that an existing LP relaxation is the same as the concave relaxation on the Lov\'{a}sz extension for the Sup-ML problem; we then prove an upper bound of $\frac{2}{k}$ on the integrality gap of the LP relaxation. These suggest that the $\frac{2}{k}$-approximation algorithm is the best possible based on the LP relaxation. For MUHV, we formulate a novel LP relaxation and prove that it is the same as the convex relaxation on the Lov\'{a}sz extension for the Sub-ML problem; we then show a lower bound of $2 - \frac{2}{k}$ on the integrality gap of the LP relaxation. Similarly, these suggest that the $(2 - \frac{2}{k})$-approximation algorithm is the best possible based on the LP relaxation. Lastly, we prove that this $(2 - \frac{2}{k})$-approximation is optimal for the MUHV problem, assuming the Unique Games Conjecture.