The optimal base of a matroid/with tree-type constraints

LetM be a matroid defined on a weighted finite setE=(e ₁, ...,e _n ).l(e) is the weight ofe∈E. P=(X ₁, ...,X _m ) is a set of subsets ofE. ?X _i ,X _j ∈P, ifX _i ∩X _j ≠ø (the empty set), then eitherX _i ?X _j orX _j ?X _i . For eachX _i ∈P, there are two associate nonnegative integersa _i andb _i witha _i ≤b _i ≤|X _i |. We call a baseT ofM a feasible base with respect toP (or simply call it a feasible base ofM), if ?X _i ∈P,a _i ≤|X _i ∩T|≤b _i . A baseT′ is called optimal if:

1. T′ is feasible,
2. \(l(T') = \sum\limits_{e \in T'} {l(e) = \min (\sum\limits_{e \in T} {l(e)|T} }\) is a feasible base ofM).

In this paper we present a polynomial algorithm to solve the optimal base problem.