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 Lovász 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 − 2 k ( 2 k , respectively). These general results imply that the MHV and the MUHV problems can also be approximated within 2 k and 2− 2 k , respectively, using the same approximation algorithms. For MHV, this 2 k -approximation algorithm improves the previous best approximation ratio max{ 1 k , 1 ∆+1 }, where ∆ 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 Lovász extension for the Sup-ML problem; we then prove an upper bound of 2 k on the integrality gap of the LP relaxation. These suggest that the 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 Lovász extension for the Sub-ML problem; we then show a lower bound of 2 − 2 k on the integrality gap of the LP relaxation. Similarly, these suggest that the (2 − 2 k )-approximation algorithm is the best possible based on the LP relaxation. Lastly, we prove that this (2 − 2 k )-approximation is optimal for the MUHV problem, assuming the Unique Games Conjecture.
ACM Subject ClassificationDummy classification -please refer to http://www.acm.org/ about/class/ccs98-html