Deconstructing intractability - A multivariate complexity analysis of interval constrained coloring

The NP-hard Interval Constrained Coloring (ICC) problem appears in the interpretation of experimental data in biochemistry dealing with protein fragments. Given a set of m integer intervals in the range 1 to n and a set of m associated multisets of colors (specifying for each interval the colors to be used for its elements), one asks whether there is a “consistent” coloring for all integer points from {1, . . . , n} that complies with the constraints specified by the color multisets. We thoroughly analyze a known NP-hardness proof for ICC. In this way, we identify numerous parameters that naturally occur in ICC and strongly influence its practical solvability. Accordingly, we present several positive (fixed-parameter) tractability results exploiting various parameterizations. We substantiate the usefulness of this “multivariate algorithmics approach” by presenting experimental results with real-world data.