In mathematics, the relaxation of a (mixed) integer linear program is the problem that arises by removing the integrality constraint of each variable. For example, in a 0–1 integer program, all constraints are of the form
x i ∈ { 0 , 1 } {\displaystyle x_{i}\in \{0,1\}} . The relaxation of the original integer program instead uses a collection of linear constraints
0 ≤ x i ≤ 1. {\displaystyle 0\leq x_{i}\leq 1.}
The resulting relaxation is a linear program, hence the name. This relaxation technique transforms an NP-hard optimization problem (integer programming) into a related problem that is solvable in polynomial time (linear programming); the solution to the relaxed linear program can be used to gain information about the solution to the original integer program.
Example Consider the set cover problem, the linear programming relaxation of which was first considered by Lovász in 1975. In this problem, one is given as input a family of sets F = {S0, S1, ...}; the task is to find a subfamily, with as few sets as possible, having the same union as F. To formulate this as a 0–1 integer program, form an indicator variable xi for each set Si, that takes the value 1 when Si belongs to the chosen subfamily and 0 when it does not. Then a valid cover can be described by an assignment of values to the indicator variables satisfying the constraints
x i ∈ { 0 , 1 } {\displaystyle \textstyle x_{i}\in \{0,1\}}
(that is, only the specified indicator variable values are allowed) and, for each element ej of the union of F,
∑ { i ∣ e j ∈ S i } x i ≥ 1 {\displaystyle \textstyle \sum _{\{i\mid e_{j}\in S_{i}\}}x_{i}\geq 1}
(that is, each element is covered). The minimum set cover corresponds to the assignment of indicator variables satisfying these constraints and minimizing the linear objective function
min ∑ i x i . {\displaystyle \textstyle \min \sum _{i}x_{i}.}
The linear programming relaxation of the set cover problem describes a fractional cover in which the input sets are assigned weights such that the total weight of the sets containing each element is at least one and the total weight of all sets is minimized. As a specific example of the set cover problem, consider the instance F = {{a, b}, {b, c}, {a, c}}. There are three optimal set covers, each of which includes two of the three given sets. Thus, the optimal value of the objective function of the corresponding 0–1 integer program is 2, the number of sets in the optimal covers. However, there is a fractional solution in which each set is assigned the weight 1/2, and for which the total value of the objective function is 3/2. Thus, in this example, the linear programming relaxation has a value 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 it happens that, in the optimal solution, all variables have integer values, then 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.
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