Many problems in bioinformatics are about finding strings that approximately represent a collection of given strings. We look at more general problems where some input strings can be classified as outliers. The Close to Most Strings problem is, given a set S of same-length strings, and a parameter d, find a string x that maximizes the number of "non-outliers" within Hamming distance d of x. We prove this problem has no PTAS unless ZPP=NP, correcting a decade-old mistake. The Most Strings with Few Bad Columns problem is to find a maximum-size subset of input strings so that the number of non-identical positions is at most k; we show it has no PTAS unless P=NP. We also observe Closest to k Strings has no EPTAS unless W=FPT. In sum, outliers help model problems associated with using biological data, but we show the problem of finding an approximate solution is computationally difficult.
Christina Boucher, Gadi Landau, Avivit Levy, David Pritchard, and Oren Weimann. On Approximating String Selection Problems with Outliers. In proceedings of the 23rd Annual Combinatorial Pattern Matching (CPM 2012), pages 427--438.