differing from that of the unrelaxed 0–1 integer program. Solution quality of relaxed and original programs The linear programming relaxation of an integer program may be solved using any standard linear programming technique. If the optimal solution to the linear program happens to have all variables either 0 or 1, it will also be an optimal solution to the original integer program. However, this is generally not true, except for some special cases (e.g., problems with totally unimodular matrix specifications.) In all cases, though, the solution quality of the linear program is at least as good as that of the integer program, because any integer program solution would also be a valid linear program solution. That is, in a maximization problem, the relaxed program has a value greater than or equal to that of the original program, while in a minimization problem such as the set cover problem the relaxed program has a value smaller than or equal to that of the original program. Thus, the relaxation provides an optimistic bound on the integer program's solution. In the example instance of the set cover problem described above, in which the relaxation has an optimal solution value of 3/2, we can deduce that the optimal solution value of the unrelaxed integer program is at least as large. Since the set cover problem has solution values that are integers (the numbers of sets chosen in the subfamily), the optimal solution quality must be at least as large as the next larger integer, 2. Thus, in this instance, despite having a different value from the unrelaxed problem, the linear programming relaxation gives us a tight lower bound on the solution quality of the original problem. Approximation and integrality gap Linear programming relaxation is a standard technique for designing approximation algorithms for hard optimization problems. In this application, an important concept is the integrality gap, the maximum ratio between the solution quality of the integer program and of its relaxation. In an instance of a minimization problem, if the real minimum (the minimum of the integer problem) is , and the relaxed minimum (the minimum of the linear programming relaxation) is , then the integrality gap of that instance is . In a maximization problem the fraction is reversed. The integrality gap is always at least 1. In the example above, the instance F = {{a, b}, {b, c}, {a, c}} shows an integrality