Non-approximability of Weighted Multiple Sequence Alignment

We consider a weighted generalization of multiple sequence alignment with sumof-pair score. Multiple sequence alignment without weights is known to be NPcomplete and can be approximated within a constant factor, but it is unknown whether it has a polynomial time approximation scheme. Weighted multiple sequence alignment can be approximated within a factor of O(log n) where n is the number of sequences. We prove that weighted multiple sequence alignment is MAX SNP-hard and give a numerical lower bound on its approximability, namely 324 323 − ǫ. This lower bound is obtained already for the simple binary weighted case where the weights are restricted to 0 and 1. Furthermore, we show that weighted multiple sequence alignment and its restriction to binary weights can be approximated to the same degree.