On the approximability of the maximum interval constrained coloring problem

In the Maximum Interval Constrained Coloring problem, we are given a set of intervals on a line and a k-dimensional requirement vector for each interval, specifying how many vertices of each of k colors should appear in the interval. The objective is to color the vertices of the line with k colors so as to maximize the total weight of intervals for which the requirement is satisfied. This \(\mathcal{NP}\)-hard combinatorial problem arises in the interpretation of data on protein structure emanating from experiments based on hydrogen/deuterium exchange and mass spectrometry. For constant k, we give a factor \(O(\sqrt{|{\textsc{Opt}}|})\)-approximation algorithm, where Opt is the smallest-cardinality maximum-weight solution. We show further that, even for k = 2, the problem remains APX-hard.