Submodular and supermodular multi-labeling, and vertex happiness.
Yao Xu, Randy G. Goebel, Guohui Lin · arXiv (Cornell University) · 2016
In this paper, we investigate the submodular multi-labeling (Sub-ML) problem, a more general version of the submodular multiway partition (Sub-MP), which captures many cut problems as special cases, including the edge-/node-weighted multiway cut and the hypergraph multiway cut. We also study the complement of Sub-ML, the supermodular multi-labeling (Sup-ML) problem. We propose a convex (concave, respectively) relaxation for the Sub-ML (Sup-ML, respectively) problem based on the Lov\'{a}sz extension, which can be solved in polynomial time. By applying a randomized rounding, we prove that Sub-ML can be approximated within a factor of $2 - \frac{2}{k}$ and Sup-ML can be approximated within a factor of $\frac{2}{k}$, where $k$ is the number of labels. In addition, we find that a recently studied vertex-coloring problem, the maximum happy vertices (MHV) problem, can be casted as a special case of Sup-ML. Hence, MHV can be approximated within $\frac{2}{k}$, which improves the previous best $\frac{1}{k}$-approximation. Based on the associated LP relaxation, we further prove that the $\frac{2}{k}$-approximation is the best possible for MHV. For the complementary minimum unhappy vertices (MUHV) problem, casted as a special case of Sub-ML, it can be approximated within $2 - \frac{2}{k}$ too; we prove that $2 - \frac{2}{k}$ is the best possible based on the associated LP relaxation; lastly, we show that a $(2 - \frac{2}{k} - \epsilon)$-approximation for MUHV is NP-hard under the Unique Games Conjecture, for any positive $\epsilon$.