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Table 1. Comparison of dual simplex and barrier for the solution of the LPs Name 1aab 1fjlA 1hfh 1hpi 1csy 1pfc 1tgxA 3cyr Dual Auto 23 1:05:34 6:06:12 11:56 1:14:58 LIMIT 7:40 42:29 Steepest Descent 20 47:58 6:33:35 11:40 1:41:54 LIMIT 7:43 45:26 Barrier 16 7:51 29:42 7:49 9:24 40:38 2:08 6:55 420 E. Althaus et al. Table 2. Integrality gaps and associated LP solution time depending on the classes of inequalities separated Name 1aab 1fjlA 1hfh 1hpi 1csy 1pfc 1tgxA 3cyr w/ all ineq. gap time 0 16 0 7:51 0 29:42 2 5:02 0 9:24 0 40:38 0 2:08 0 6:55 w/o (10) gap time 0 16 0 7:51 2 22:38 10 2:19 0 9:24 0 40:38 0 2:08 0 6:55 w/o (9), (10) gap time 25 14 71 8:49 158 9:44 52 1:34 138 5:48 101 12:44 169 1:18 99 6:17 w/o (8), (10) gap time 68 31 114 10:01 399 15:55 154 1:14 353 6:55 234 8:53 144 2:27 228 4:23 w/o (8), (9), (10) gap time 111 2 206 34 521 42 195 5 433 22 367 47 202 4 284 15 Separation After extensive experiments, we selected the following separation strategy.

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 }. After having computed the above values, for every alignment variable xe correspondj ing to edge e = {vli , vm }, we let Uxe = σ ij + π ij (l − 1, m − 1) + we + ωij (l + 1, m + 1), and perform the associated reduction of the alignment variables in O(n2 ) time, whereas i )j , we let for every gap variable ya corresponding to gap arc a = (vli , vm Uya = σ ij + max (π ij (l − 1, p) + ωij (m + 1, p + 1)).

K, i < j in O( k−1 j =i+1 |s ||s | max{|s |, |s |}) = O(n ) time by the foli=1 lowing modification of the standard dynamic programming technique for the case in which wa = 0 for a ∈ Ag [15]. As already done, we implicitly represent the dynamic programming recursion by formulating the problem on a directed acyclic graph Qij having a node n{v i ,v j } for i l m A branch-and-cut algorithm for multiple sequence alignment 417 l = 1, . . , |s i | + 1 and m = 1, . . , |s j | + 1. For l = 1, . .

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