By Ghiani G., Laporte G.

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Extra resources for A Branch-and-cut Algorithm for the Undirected Rural Postman Problem

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More precisely, the notation 0(6) means that the complexity has an upper limit set by a linear function in b whereas the notation 0(6) enables to specify that the complexity is equivalent to 6. The simplification by the notation 0(6) may lead to paradoxical situations. In effect, an algorithm A in 0{Max'^) may be slower than an algorithm B in 0(2^^^) for certain problems. Let us take, for example a scheduling problem where Max is the number of jobs n and let us suppose that the complexity of the algorithm A is 2^^°^n^ and that of the algorithm B is 2'^.

P = ^ . Thus, the average complexity is equal to p(l + 2 + 3 + ... + n) = \^ = ^ ^ . We notice that the calculation is only valid if we are sure that the element belongs to the list. In the opposite case, the calculation of the average complexity is even more complicated as it causes the law of generation of the elements of the list to intervene. Search for an element belonging to a list /* elt the searched element */ / * n is the list size */ /* list is the list of elements */ /* We assume that elt belongs to the list */ i = 0; While {elt ^ list\i]) Do z = i-h 1; End While; Fig.

If 1. yr instance of Do', 3 / an instance of Do such that Sj C Sp. 2. I can be constructed in polynomial time starting with V. then a polynomial Turing reduction exists such that O' OCT O. Considering strong A/'T^-hardness, similar results to those for proving strong A^'P-completeness have to be stated. Thus polynomial Turing reductions are not sufficient to prove strong A/'P-hardness, and similarly the notion of pseudo-polynomial Turing reduction must be considered. A/'T^-hardness is definitely a general complexity state, an AfV-hdud problem being at least as hard as any problem in class ÄfV.

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