Lecture 13
نویسندگان
چکیده
We discuss two ways to determine, given I ⊂ S, whether I ∈ I. The first method is based on the previous lecture, where we considered the set Ŝ = ∪̇Ŝi, a union of disjoint copies of the Si. Formally, we write Ŝ = {(e, i) : e ∈ Si}. There is a natural mapping f : Ŝ → ∪Si which maps (e, i) to e. We can now define a partition matroid Mp over the ground set Ŝ, where I(Mp) = { I ⊆ Ŝ s.t. ∀e ∈ S : |I ∩ {(e, i) : e ∈ Si}| ≤ 1 } . (Thus, an independent set I in Mp does not contain both (e, i) and (e, j) for any e ∈ S, i 6= j.) We can now test for independence in the matroid union by looking at f−1(U) and using our matroid intersection algorithm with the partition matroid to look for a maximal such subset of f−1(U). A potential difficulty of this approach is that Ŝ can be large. (If, for example, all Mi are defined over the same ground set, then Ŝ is k times as large as this ground set.)
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