Approximability And Inapproximability For Maximum K-Edge-Colored Clustering Problem

We consider the Max k-Edge-Colored Clustering problem (abbreviated as MAX-k-EC). We are given an edge-colored graph with k colors. Each edge of the graph has a positive weight. We seek to color the vertices of the graph so as to maximize the total weight of the edges which have the same color at their extremities. The problem was introduced by Angel et al. [2]. We give a polynomial-time algorithm for MAX-k-EC with an approximation factor 49/144 approximate to 0.34, which significantly improves the best previously known factor 7/23 approximate to 0.304, obtained by Ageev and Kononov [I]. We also present an upper bound of 241/248 approximate to 0.972 on the inapproximability of MAX-k-EC. This is the first inapproximability result for this problem.