Tight polynominal bounds for matroidal knapsacks
Let F be the independence family of a matroid on N = {l_brace}1, ..., n{r_brace} and let B denote the set of the incidence vectors of its bases. The Matroidal Knapsack Problem is the problem of finding max (cx : x {element_of} P) where P = {l_brace}x {element_of} {l_brace}0, 1{r_brace}{sup n} : ax {<=} b, x {element_of} B{r_brace}, a = A{sub 1}, ..., a{sub n} is a nonnegative integer vector and b {>=} is an integer. Problems belonging to this class, such as the Multiple Choice Knapsack or the Capacitated Minimal Spanning Tree, are known to be NP-hard. However, there are some instances that are easy to solve, say when (x {element_of} (0, 1){sup n} : ax {<=} b) is a matroid on N. If it is not the case we will show how to compute polynomially an upper bound of z: we begin by proving that there exists a valid inequality for P which is a matroidal relaxation for the knapsack constraint. We add this constraint to P, in order to build a strong Lagrangean Dual. Then we show that, using a polyhedral description for the Matroidal Knapsack that this bound strictly dominates z{sub lp} and it can be obtained in polynomial time.
- OSTI ID:
- 35762
- Report Number(s):
- CONF-9408161--
- Country of Publication:
- United States
- Language:
- English
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