Noisy Data Make the Partial Digest Problem NP - hardTECHNICAL

The Partial Digest problem { well-known for its applications in computational biology and for the intriguingly open status of its computational complexity { asks for the coordinates of n points on a line such that the pairwise distances of the points form a given multi-set of ? n 2 distances. In an eeort to model real-life data, we study the computational complexity of a minimization version of Partial Digest, in which only a subset of all pairwise distances is given and the rest are lacking due to experimental errors. We show that this variation is NP-hard to solve exactly, thus making the existence of polynomial-time algorithms for this problem extremely unlikely. Our result answers an open question posed by Pevzner (2000). We then study a maximiza-tion version of Partial Digest where a superset of all pairwise distances is given, with some additional distances due to inaccurate measurements. We show that this maximization version is NP-hard to approximate to within a factor of jDj 1 2 ? for any > 0, where jDj is the number of input distances, which implies that polynomial-time algorithms cannot even guarantee to nd a solution for the problem that comes close to the optimum. Our inapproximability result is tight up to low-order terms as we give a trivial approximation algorithm that achieves a matching approximation ratio. Our optimization variations model two diierent error types that occur in real-life data.