By Ivanyi A. (ed.)
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Additional info for Algorithms of informatics, vol. 3
In that case the same constraint is stored many times, because a branch on a higher level may have many subbranches. As matter of fact the number of branches is very high in the case of large scale problems, thus the memory required by this solution is very high. The other option is that only those informations are stored which are necessary to the complete reconstruction of the branch. e. the branch from which it was generated directly, bound of the objective function on the branch, index of the branching variable, branch defining constraint of the branching variable.
N . 78) j=1,j=i The second part is very similar. It claims that the salesman must arrive to each city from somewhere else again exactly once: n xij = 1 j = 1, . . , n . 79) are the constraints of an assignment problem. 79) is really an assignment problem. They don’t exclude solutions consisting of several smaller tours. For example if n = 6 and x12 = x23 = x31 = 1 and x45 = x56 = x64 = 1 then all other variables must be zero. The solution consists of two smaller tours. The first one visits only cities 1, 2, and 3, the second one goes through the cities 4, 5, and 6.
Its main tool is the cut which is based on the observation that possible to determine linear inequalities such that they cut the non-integer optimal solution of the current LP relaxation, but they do not cut any integer feasible solution. A systematic generation of cuts leads to a finite algorithm which finds an optimal solution and proves its optimality if optimal solution exist, otherwise it proves the non-existence of the optimal solution. From geometrical point of view the result of the introducing of the cuts is that the shape of the polyhedral set of the last LP relaxation is very similar to the integer hull in the neighborhood of the optimal solution.