By Ivanyi A. (ed.)

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Hence the inequality ( dpj + fj ) fp ≤ fp fj − fj must hold. It can be reduced to It is true as dpj ≤ −1 and dpj fp ≤ −fj . −1 ≤ −fj < 0. 61) the term d0j (1 − fj )2 fp (1 − fj ) is present among the terms of the minimum of the upper branch. 57). 4. Branching strategies The B&B frame doesn’t have any restriction in the selection of the unfathomed node for the next branching in row 7 of Branch-and-Bound. First two extreme strategies are discussed with pros and cons. The same considerations have to be taken in almost all applications of B&B.

E. the optimal integer solution of the branch as it will be seen in the next section, too. Right now assume that the integer optimal solution with objective function value 4 is uncovered. At this stage of the algorithm the only unfathomed branch is Branch 4 with feasible region OHG. Obviously the optimal solution is point G=( 53 ,1). Its objective function value is 13 3 . Thus it cannot contain a better feasible solution than the known (1,2). Hence the algorithm is finished. 2. The linear programming background of the method The first ever general method solving linear programming problems were discovered by George Dantzig and called simplex method.

N}, 1 ≤ |S| ≤ n − 1 : i∈S j∈Sj=i xij ≤ |S| . 6. Branch and Cut 1259 The other logic claims that the salesman must leave all such sets. Let S¯ = {1, 2, . . , n} \ S. Then the subtour elimination constraints are the inequalities ∀ S ⊂ {1, 2, . . , n}, 1 ≤ |S| ≤ n − 1 : i∈S j∈S¯ xij ≥ 1 . 81) The numbers of the two types of constraints are equal and exponential. 81) are satisfied by only binary vectors being characteristic vectors of complete tours but the polyhedral set of the LP relaxation is strictly larger than the integer hull.

### Algorithms of informatics, vol. 3 by Ivanyi A. (ed.)

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