By Vandenbussche D., Nemhauser G. L.

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Now look at the two ‘recursive calls,’ which really aren’t quite. The first one of them sorts the array to the left of xi . That is indeed a recursive call, because we can just change the ‘n’ to ‘i − 1’ and call Quicksort. The second recursive call is the problem. It wants to sort a portion of the array that doesn’t begin at the beginning of the array. The routine Quicksort as written so far doesn’t have enough flexibility to do that. So we will have to give it some more parameters. Instead of trying to sort all of the given array, we will write a routine that sorts only the portion of the given array x that extends from x[lef t] to x[right], inclusive, where lef t and right are input parameters.

In Fig. 4(b) we show the same graph with a successful proper coloring of its vertices in 4 colors. Fig. 4(a) Fig. 4(b) The chromatic number χ(G) of a graph G is the minimum number of colors that can be used in a proper coloring of the vertices of G. , it is a graph that can be 2-colored. That means that the vertices of a bipartite graph can be divided into two classes ‘R’ and ‘Y’ such that no edge of the graph runs between two ‘R’ vertices or between two ‘Y’ vertices. Bipartite graphs are most often drawn, as in Fig.

For instance, if we are given {9, 4, 7, 2, 1}, then we want our program to output the sorted array {1, 2, 4, 7, 9}. There are many methods of sorting, but we are going to concentrate on methods that rely on only two kinds of basic operations, called comparisons and interchanges. This means that our sorting routine is allowed to (a) pick up two numbers (‘keys’) from the array, compare them, and decide which is larger. (b) interchange the positions of two selected keys. Here is an example of a rather primitive sorting algorithm: (i) find, by successive comparisons, the smallest key (ii) interchange it with the first key (iii) find the second smallest key (iv) interchange it with the second key, etc.

### A branch-and-cut algorithm for nonconvex quadratic programs with box constraints by Vandenbussche D., Nemhauser G. L.

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