A polynomial-delay algorithm for enumerating approximate solutions to the interval constrained coloring problem

Abstract

We study the interval constrained coloring problem, a combinatorial problem arising in the interpretation of data on protein structure emanating from experiments based on hydrogen/deuterium exchange and mass spectrometry. The problem captures the challenging task of increasing the spatial resolution of experimental data in order to get a better picture of the protein structure. Since solutions proposed by any algorithmic framework have to ultimately be verified by biochemists, it is important to provide not just a single solution, but a valuable set of candidate solutions. Our contribution is a polynomial-delay, polynomial-space algorithm for enumerating all exact solutions plus further approximate solutions, which are guaranteed to be within an absolute error of two of the optimum within fragments of the protein, that is, within sets of consecutive residues. Our experiments indicate that the quality of the approximate solutions is comparable to the optimal ones in terms of deviation from the underlying true solution. In addition, the experiments also confirm the effectiveness of the method in reducing the delay between two consecutive solutions considerably, compared to what it takes an integer programming solver to produce the next exact solution.