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section20

# section20 - 20 Solving integer programs We now have a...

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20 Solving integer programs We now have a systematic understanding of how to solve linear programs. However, we have so far developed no ideas about how to solve the kinds of integer programs that we encountered earlier. The only exception is the special case of transportation problems, where the integrality restriction on the variables turns out to be essentially no restriction at all. Unlike trans- portation problems (and a few other special problem classes), most integer programs are much harder to solve than linear programs. In this section we will start to outline a general computational approach. Consider the following example of an integer program with binary vari- ables. maximize 3 x 1 + 2 x 2 + 4 x 3 subject to x 1 + 2 x 2 + x 3 1 x 1 + 2 x 2 + x 3 2 x 1 + x 2 - 2 x 3 0 x 1 , x 2 , x 3 { 0 , 1 } . One foolproof approach is complete enumeration : we simply list all the pos- sible solutions, check which are feasible, and among these, find the best. In this case, we arrive at the following enumeration. x 1 x 2 x 3 Feasible Objective 0 0 0 no 0 0 1 no 0 1 0 yes 2 0 1 1 no 1 0 0 yes 3 1 0 1 no 1 1 0 no 1 1 1 no From the table, we see that the optimal solution of the integer program is [1 , 0 , 0] T . For this small example, complete enumeration was simple and effective. In general, however, this approach is clearly not practical for anything but the smallest problems, because the number of solutions we must enumerate grows exponentially in the number of binary variables. The next natural idea we might try is rounding . We could use the sim- plex method to solve the linear programming relaxation : that is, the linear 108

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program we obtain by dropping all the integrality restrictions. We might hope that the optimal solution to the integer program is somehow “close” to the optimal solution of the relaxation. Unfortunately, simple examples show that rounding may fail to say any- thing helpful about the integer program. Consider, for example, the simple integer program maximize x 2 subject to - 2 x 1 + 2 x 2 1 2 x 1 + 2 x 2 7 x 1 , x 2 0 , integer . The optimal solution of the linear programming relaxation is [ 3 2 , 1] T : round- ing the fractional component either up or down leads to an infeasible solution. Rounding gives us no obvious information about optimal solutions of the in- teger program (in this case [1 , 1] T and [2 , 1] T ).
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