  We consider the problem Minimum Error Correction (MEC). A MEC instance is an
n x m matrix M with entries from {0,1,-}. Feasible solutions are composed of
two binary m-bit strings, together with an assignment of each row of M to one
of the two strings. The objective is to minimize the number of mismatches
(errors) where the row has a value that differs from the assigned solution
string. The symbol "-" is a wildcard that matches both 0 and 1. A MEC instance
is gapless, if in each row of M all binary entries are consecutive.
  Gapless-MEC is a relevant problem in computational biology, and it is closely
related to segmentation problems that were introduced by
[Kleinberg-Papadimitriou-Raghavan STOC'98] in the context of data mining.
  Without restrictions, it is known to be UG-hard to compute an
O(1)-approximate solution to MEC. For both MEC and Gapless-MEC, the best
polynomial time approximation algorithm has a logarithmic performance
guarantee. We partially settle the approximation status of Gapless-MEC by
providing a quasi-polynomial time approximation scheme (QPTAS). Additionally,
for the relevant case where the binary part of a row is not contained in the
binary part of another row, we provide a polynomial time approximation scheme
(PTAS).
