We investigate the maximum happy vertices (MHV) problem and its complement, the minimum unhappy vertices (MUHV) problem. In order to design better approximation algorithms, we introduce the supermodular and submodular multi-labeling (Sup-ML and Sub-ML) problems and show that MHV and MUHV are special cases of Sup-ML and Sub-ML, respectively, by rewriting the objective functions as set functions. The convex relaxation on the Lovasz extension, originally presented for the submodular multi-partitioning problem, can be extended for the Sub-ML problem, thereby proving that Sub-ML (Sup-ML, respectively) can be approximated within a factor of $$2 - 2/k$$ (2 / k, respectively), where k is the number of labels. These general results imply that MHV and MUHV can also be approximated within factors of 2 / k and $$2 - 2/k$$ , respectively, using the same approximation algorithms. For the MUHV problem, we also show that it is approximation-equivalent to the hypergraph multiway cut problem; thus, MUHV is Unique Games-hard to achieve a $$(2 - 2/k - \varepsilon )$$ -approximation, for any $$\varepsilon > 0$$ . For the MHV problem, the 2 / k-approximation improves the previous best approximation ratio $$\max \{1/k, 1/\big (\varDelta + 1/g(\varDelta )\big )\}$$ , where $$\varDelta $$ is the maximum vertex degree of the input graph and $$g(\varDelta ) = (\sqrt{\varDelta } + \sqrt{\varDelta + 1})^2 \varDelta > 4 \varDelta ^2$$ . We also show that an existing LP relaxation for MHV is the same as the concave relaxation on the Lovasz extension for Sup-ML; we then prove an upper bound of 2 / k on the integrality gap of this LP relaxation, which suggests that the 2 / k-approximation is the best possible based on this LP relaxation. Lastly, we prove that it is Unique Games-hard to approximate the MHV problem within a factor of $$\varOmega (\log ^2 k / k)$$ .