Clustering on k -edge-colored graphs

We study the Max k -colored clustering problem, where given an edge-colored graph with k colors, we seek to color the vertices of the graph so as to find a clustering of the vertices maximizing the number (or the weight) of matched edges, i.e.źthe edges having the same color as their extremities. We show that the cardinality problem is NP-hard even for edge-colored bipartite graphs with a chromatic degree equal to two and k ź 3 . Our main result is a constant approximation algorithm for the weighted version of the Max k -colored clustering problem which is based on a rounding of a natural linear programming relaxation. For graphs with chromatic degree equal to two we improve this ratio by exploiting the relation of our problem with the Max 2-and problem. We also present a reduction to the maximum-weight independent set (IS) problem in bipartite graphs which leads to a polynomial time algorithm for the case of two colors.