Improved Approximations for the Max k-Colored Clustering Problem

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.