Algorithms And Data Structures

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By Althaus E.

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Variable reduction Even for rather small instances, the number of variables in our ILP k i j 2 formulation is very large, namely there are k−1 j =i+1 |s ||s | = O(n ) edge varii=1 ables and ki=1 (k − 1) |s 2|+1 = O(n3 ) arc variables. For example, an instance with 5 strings of length 100 has 201, 000 variables. Most of these variables are very unlikely to take the value 1 in an optimal solution. In this section, we describe a simple but successful method to reduce the number of variables. Assume we know a lower bound L on the optimum, typically found by a heuristic.

All computing times in the tables are given in the format “hh:mm:ss”. 1. Initial experiments In this section we discuss and justify experimentally some key choices in our implementation. Solution of the LPs As the solution of the LPs is the computational bottleneck of the approach, we tried different methods. For the easy instances, reoptimizing the LPs with the dual simplex algorithm was slightly faster than solving them from scratch with the barrier method. On the other hand, contrary to what is generally observed in other cases, for the challenging instances, the barrier method widely outperformed the simplex algorithm, provided one resigned to crossover to a basic solution.

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, . . , k; i < j, , i, j = 1, . . , k; i < j, ≥ z{v i ,v j } + x{v i l j l+1 ,vm } m , l = 1, . . , |s i | − 1; m = 1, .

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