08-matroids - Algorithms Lecture 8: Matroids [ Fa10 ] The...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Algorithms Lecture 8: Matroids [ Fa10 ] 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 8 Matroids ? 8.1 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 vectorsmatroid 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 Whitneys 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....
View Full Document

Page1 / 6

08-matroids - Algorithms Lecture 8: Matroids [ Fa10 ] The...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online