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:
“Give a scientist a problem and he will probably provide a solution; historians and sociologists, by contrast, can offer only opinions. Ask a dozen chemists the composition of an organic compound such as methane, and within a short time all twelve will have come up with the same solution of CH4. Ask, however, a dozen economists or sociologists to provide policies to reduce unemployment or the level of crime and twelve widely differing opinions are likely to be offered.”
—Derek Gjertsen, British scientist, author. Science and Philosophy: Past and Present, ch. 3, Penguin (1989)
“The best quality tea must have creases like the leathern boot of Tartar horsemen, curl like the dewlap of a mighty bullock, unfold like a mist rising out of a ravine, gleam like a lake touched by a zephyr, and be wet and soft like a fine earth newly swept by rain.”
—Lu Yu (d. 804)
“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)
“If we remembered everything, we should on most occasions be as ill off as if we remembered nothing. It would take us as long to recall a space of time as it took the original time to elapse, and we should never get ahead with our thinking. All recollected times undergo, accordingly, what M. Ribot calls foreshortening; and this foreshortening is due to the omission of an enormous number of facts which filled them.”
—William James (1842–1910)
“Government ... thought [it] could transform the country through massive national programs, but often the programs did not work. Too often they only made things worse. In our rush to accomplish great deeds quickly, we trampled on sound principles of restraint and endangered the rights of individuals.”
—Gerald R. Ford (b. 1913)