Singular Manual: Integer programming

| ### C.6.4 Integer programming (1) Compute the toric ideal I(A) using one of the algorithms in the previous section. (2) Compute the reduced Groebner basis G(c) of I(A) w.r.t. . (3) Reduce modulo G(c) using the Hironaka division algorithm. If the result of this reduction is , then is an optimal solution of the given instance. If no initial solution is known, we are nevertheless able to solve the problem with similar techniques. For this purpose we replace our instance by an extended instance with the matrix used in the Conti-Traverso algorithm. Indeed, the Conti-Traverso algorithm offers the possibility to verify solvability of a given instance and to find an initial solution in the case of existence (but none of the other algorithms does!). Details can be found in see [CoTr91] and see [The99]. An implementation of the above algorithm and some examples can be found in intprog_lib. In general, classical methods for solving IP instances like Branch-and-Bound methods seem to be faster than the methods using toric ideals. But the latter have one great advantage: If one wants to solve various instances that differ only by the vector , one has to perform steps (1) and (2) above only once. As the running time of step (3) is very short, solving all the instances is not much harder than solving one single instance. For a detailed discussion see see [The99].

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