  Sequence alignment supports numerous tasks in bioinformatics, natural
language processing, pattern recognition, social sciences, and others fields.
While the alignment of two sequences may be performed swiftly in many
applications, the simultaneous alignment of multiple sequences proved to be
naturally more intricate. Although most multiple sequence alignment (MSA)
formulations are NP-hard, several approaches have been developed, as they can
outperform pairwise alignment methods or are necessary for some applications.
  Taking into account not only similarities but also the lengths of the
compared sequences (i.e. normalization) can provide better alignment results
than both unnormalized or post-normalized approaches. While some normalized
methods have been developed for pairwise sequence alignment, none have been
proposed for MSA. This work is a first effort towards the development of
normalized methods for MSA.
  We discuss multiple aspects of normalized multiple sequence alignment (NMSA).
We define three new criteria for computing normalized scores when aligning
multiple sequences, showing the NP-hardness and exact algorithms for solving
the NMSA using those criteria. In addition, we provide approximation algorithms
for MSA and NMSA for some classes of scoring matrices.

Data Structures and Algorithms