By Althaus E.

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**Extra info for A branch-and-cut algorithm for multiple sequence alignment**

**Example text**

Sli of s i with substring s1 . . sm of s j , for l = 1, . . , |s i | and m = 1, . . , |s j |. Moreover, in a completely analogous way and in the same running time one can find the value ωij (l, m) of the optimal alignment of substring sli . . s|si i | of s i with substring j j sm . . s|s j | of s j . The sum of the values of all the pairwise optimal alignments is clearly an upper bound on the value of the optimal alignment, called the pairwise upper bound. For convenience, let σ ij denote the sum of the values of the optimal alignments between all string pairs different from {s i , s j }.

In accordance with the results shown, in our implementation we use the barrier method without crossover for the solution of each LP. Furthermore, after each LP solution, we remove all inequalities whose associated dual variable is zero, since, although the total number of LPs to be solved increased slightly with this modification, the overall solution time decreased. As the number of variables was quite large, we also tried to apply various pricing policies, including pricing based on a relaxation of the problem (in our case, the relaxation associated with the pairwise upper bound) following the approach described in [10].

Dual of the customary LP formulation of longest path on a directed acyclic graph, using arc variables) is more than 1 − y ∗ i i j : (v1 ,v ) |s i | min z{v i j |s i | ,v1 } subject to z{v i ,v j } = x ∗ i 1 z{v i j l+1 ,vm } z{v i ,v j m−1 } l j } |s j | {v1 ,v |s j | , ≥ z{v i ,v j } + x ∗ i ≥ z{v i ,v j } + x ∗ i j , l = 1, . . , |s i | − 1; m = 1, . . , |s j |, {vl ,vm−1 } m l j {vl+1 ,vm } m l , l = 1, . . , |s i |; m = 2, . . , |s j |. Accordingly, in order to enforce that all maximal clique inequalities without gap variables j are satisfied in the LP relaxation we consider, we add variables z{v i ,v j } for {vli , vm } ∈ E l m along with the following constraints: z{v i j |s i | ,v1 } z{v i ,v j 1 |s j | } z{v i j l+1 ,vm } ≤ 1 − y(v i ,v i 1 = x{v i ,v j 1 |s j | } |s i | )j , i, j = 1, .