Steiner problem in graphs: Lagrangean relaxation and strong valid inequalities
We study the use of strong valid inequalities, within a Lagrangean based Branch-and-Bound algorithm, for the Steiner Problem in Graphs. The approach is, in many respects, similar to Branch-and-Cut algorithms in Linear Programming. Inequalities are explicitly used only when they first become violated at the solution of a Lagrangean problem. At that stage they are dualized in a Lagrangean fashion. Inequalities are discarded whenever their associated Lagrangean multipliers become equal to zero. Proceeding in this way we are able to deal, in a tractable manner, with valid inequalities that are exponential in number. The rationale behind the approach is that, by introducing those strong valid inequalities, a better approximation of the convex hull of integer solutions, for the associated Linear Programming relaxation, is achieved. Since Lagrangean bounds could be, theoretically speaking, at least as sharp as those Linear Programming bounds, we hope to benefit. Computational experiments show that our algorithm improves substantially on the results of a straightforward, more traditional, Lagrangean Relaxation implementation. Indeed, we managed to close substantial duality gaps (sometimes of, otherwise, as much as 30%). Our algorithm is also competitive, in terms of CPU time, with Branch-and-Cut type algorithms, for the test problems we consider. Here, it should be emphasized that our Lagrangean equivalent to the Separation Problem is solved in linear time as opposed to O(n{sup 4}) for the general case. This is due to the underlying structure of our Lagrangean relaxation solutions, which are Spanning Trees. This favourable situation is not particular to the Steiner Problem in Graphs and would prevail for other problems as well.
- OSTI ID:
- 36244
- Report Number(s):
- CONF-9408161--
- Country of Publication:
- United States
- Language:
- English
Similar Records
Sequence of polyhedral relaxations for nonlinear univariate functions
On-line generalized Steiner problem