Approximation Algorithms for Vertex Happiness

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 Lovász 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- ε) -approximation, for any ε> 0. For the MHV problem, the 2 / k-approximation improves the previous best approximation ratio max { 1 / k, 1 / (Δ+ 1 / g(Δ)) } , where Δ is the maximum vertex degree of the input graph and g(Δ)=(Δ+Δ+1)2Δ>4Δ2. We also show that an existing LP relaxation for MHV is the same as the concave relaxation on the Lovász 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 Ω(log ²k/ k).