Instructor : Chandra Chekuri Scribe : Sreeram Kannan 1 Maximum Weight Independent Set in a Matroid , Greedy Algo - rithm , Independence and Base Polytopes
نویسنده
چکیده
We saw the definition of base, circuit, rank, span and flat of matroids last lecture. We begin this lecture by studying some more basic properties of a matroid. Exercise 1 Show that a set I ⊆ S is independent in a matroid M iff ∀y ∈ I, there exists a flat F such that I − y ⊆ F and y ∈ F. Exercise 3 Verify that (S, I *) is indeed a matroid. Remark 4 Having an independence or rank oracle for M implies one has it for M * too. Hint: Use the third property from above. 1.2.2 Deletion Definition 6 Given a matroid M = (S, I) and e ∈ S, deleting e from M generates a new matroid M = M\e = (S − e, I), (1) where I = {I − e|I ∈ I}. For Z ⊆ S, the matroid M\Z is obtained similarly by restricting the matroid M to S\Z.
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