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 a bound on the integer program's solution quality.
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.
Read more about this topic: Linear Programming Relaxation
Famous quotes containing the words solution, quality, relaxed, original and/or programs:
“To the questions of the officiously meddling police Falter replied absently and tersely; but, when he finally grew tired of this pestering, he pointed out that, having accidentally solved “the riddle of the universe,” he had yielded to artful exhortation and shared that solution with his inquisitive interlocutor, whereupon the latter had died of astonishment.”
—Vladimir Nabokov (1899–1977)
“The irregular and intimate quality of things made entirely by the human hand.”
—Willa Cather (1873–1947)
“Abba, dark death is the breaking of a glass.
The dazzled flakes and splinters disappear.
The seal is as relaxed as dirt, perdu.”
—Wallace Stevens (1879–1955)
“You have original artworks hanging on the walls oh I said edit”
—John Ashbery (b. 1927)
“[The Republicans] offer ... a detailed agenda for national renewal.... [On] reducing illegitimacy ... the state will use ... funds for programs to reduce out-of-wedlock pregnancies, to promote adoption, to establish and operate children’s group homes, to establish and operate residential group homes for unwed mothers, or for any purpose the state deems appropriate. None of the taxpayer funds may be used for abortion services or abortion counseling.”
—Newt Gingrich (b. 1943)