Approximating the interval constrained coloring problem

We consider the interval constrained coloring problem, which appears in the interpretation of experimental data in biochemistry. Monitoring hydrogen-deuterium exchange rates via mass spectroscopy experiments is a method used to obtain information about protein tertiary structure. The output of these experiments provides data about the exchange rate of residues in overlapping segments of the protein backbone. These segments must be re-assembled in order to obtain a global picture of the protein structure. The interval constrained coloring problem is the mathematical abstraction of this re-assembly process. The objective of the interval constrained coloring problem is to assign a color (exchange rate) to a set of integers (protein residues) such that a set of constraints is satisfied. Each constraint is made up of a closed interval (protein segment) and requirements on the number of elements that belong to each color class (exchange rates observed in the experiments). We show that the problem is NP-complete for arbitrary number of colors and we provide algorithms that given a feasible instance find a coloring that satisfies all the coloring requirements within ±1 of the prescribed value. In light of our first result, this is essentially the best one can hope for. Our approach is based on polyhedral theory and randomized rounding techniques. Furthermore, we develop a quasi-polynomial-time approximation scheme for a variant of our problem where we are asked to find a coloring satisfying as many fragments as possible.