Improved Approximations for the Max k-Colored Clustering Problem

Abstract

In the Max k-Colored Clustering Problem we are given an undirected graph $G = (V,E)$. Each edge $e$ of $G$ has a nonnegative weight $w(e)$ and a color $c(e)\in \mathcal{C}=\{1, 2,\ldots , k \}$. It is required to assign a color from $\mathcal{C}$ to each vertex of $G$ so as to maximize the total weight of edges whose both endpoints have the same color as the color of the edge. Angel et al. [1] show that the problem is strongly NP-hard and present a randomized constant-factor approximation algorithm for solving it. We improve this result in two directions. First, we give a more careful analysis of the algorithm in [1], which significantly improves on its approximation bound ($0.25$ instead of $1/e^2 \approx 0.135$). Second, we present a different algorithm with a better worst case performance guarantee of $7/23 \approx 0.304$. Both algorithms are based on using similar randomized rounding schemes for a natural LP relaxation of the problem. They can be derandomized in a standard way by computing conditional expectations for some estimate function.