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04d-matroids - Algorithms Non-Lecture D Matroids The...

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Algorithms Non-Lecture D: Matroids The problem is that we attempt to solve the simplest questions cleverly, thereby rendering them unusually complex. One should seek the simple solution. — Anton Pavlovich Chekhov (c. 1890) I love deadlines. I like the whooshing sound they make as they fly by. — Douglas Adams D Matroids ? D.1 Definitions Many problems that can be correctly solved by greedy algorithms can be described in terms of an abstract combinatorial object called a matroid . Matroids were first described in 1935 by the mathematician Hassler Whitney as a combinatorial generalization of linear independence of vectors—‘matroid’ means ‘something sort of like a matrix’. A matroid M is a finite collection of finite sets that satisfies three axioms: Non-emptiness: The empty set is in M . (Thus, M is not itself empty.) Heredity: If a set X is an element of M , then any subset of X is also in M . Exchange: If X and Y are two sets in M and | X | > | Y | , then there is an element x X \ Y such that Y ∪ { x } is in M . The sets in M are typically called independent sets ; for example, we would say that any subset of an independent set is independent. The union of all sets in M is called the ground set . An independent set is called a basis if it is not a proper subset of another independent set. The exchange property implies that every basis of a matroid has the same cardinality. The rank of a subset X of the ground set is the size of the largest independent subset of X . A subset of the ground set that is not in M is called dependent (surprise, surprise). Finally, a dependent set is called a circuit if every proper subset is independent. Most of this terminology is justified by Whitney’s original example: Linear matroid: Let A be any n × m matrix. A subset I ⊆ { 1 , 2 ,..., n } is independent if and only if the corresponding subset of columns of A is linearly independent. The heredity property follows directly from the definition of linear independence; the exchange property is implied by an easy dimensionality argument. A basis in any linear matroid is also a basis (in the linear-algebra sense) of the vector space spanned by the columns of A . Similarly, the rank of a set of indices is precisely the rank (in the linear-algebra sense) of the corresponding set of column vectors. Here are several other examples of matroids; some of these we will see again later. I will leave the proofs that these are actually matroids as exercises for the reader. Uniform matroid U k , n : A subset X ⊆ { 1 , 2 ,..., n } is independent if and only if | X | ≤ k . Any subset of { 1,2,..., n } of size k is a basis; any subset of size k + 1 is a circuit. Graphic / cycle matroid M ( G ) : Let G = ( V , E ) be an arbitrary undirected graph. A subset of E is independent if it defines an acyclic subgraph of G . A basis in the graphic matroid is a spanning tree of G ; a circuit in this matroid is a cycle in G .
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