The.Algorithm.Design.Manual.Springer-Verlag.1998

Integer Programming
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Integer Programming
As discussed in Section , integer programming is a fundamental combinatorial optimization problem.
It is best thought of as linear programming with the variables restricted to take only integer (instead of
real) values.
Input: A set V of integer variables, a set of inequalities over V, a maximization function f(V), and an
integer B.
Output: Does there exist an assignment of integers to V such that all inequalities are true and ?
Consider the following two examples. Suppose
A solution to this would be , . Not all problems have realizable solutions, however. For the
following problem:
the maximum value of f(v) is (given the constraints), and so there can be no solution to the
associated decision problem.
We show that integer programming is hard using a reduction from 3-SAT. For this particular reduction,
general satisfiability would work just as well, although usually 3-SAT makes reductions easier.
In which direction must the reduction go? We want to prove integer programming that is hard, and we
know that 3-SAT is hard. If I could solve 3-SAT using integer programming and integer programming
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