Topics in Combinatorial Optimization Orlando Lee – Unicamp 28 de maio de 2014 Orlando Lee – Unicamp Topics in Combinatorial Optimization
Agradecimentos Este conjunto de slides foram preparados originalmente para o curso T´ opicos de Otimiza¸ c˜ ao Combinat´ oria no primeiro semestre de 2014 no Instituto de Computa¸ c˜ ao da Unicamp. Preparei os slides em inglˆ es simplesmente porque me deu vontade, mas as aulas ser˜ ao em portuguˆ es (do Brasil)! Agradecimentos especiais ao Prof. M´ ario Leston Rey. Sem sua ajuda, certamente estes slides nunca ficariam prontos a tempo. Qualquer erro encontrado nestes slide ´ e de minha inteira responsabilidade (Orlando Lee, 2014). Orlando Lee – Unicamp Topics in Combinatorial Optimization
Independent sets A matroid is a pair M = ( E , I ) in which I ⊆ 2 E that satisfies the following properties: (I1) ∅ ∈ I . (I2) If I ∈ I and I ′ ⊆ I , then I ′ ∈ I . (I3) If I 1 , I 2 ∈ I and | I 1 | < | I 2 | , then there exists e ∈ I 2 − I 1 such that I 1 ∪ { e } ∈ I . (Independence augmenting axiom) We say that the members of I are the independent sets of M . Orlando Lee – Unicamp Topics in Combinatorial Optimization
Rank function of a matroid In linear algebra we have important concepts such as dimension of a vector space (in particular, rank of a matrix), span of a set of vectors and bases. They all can be presented in matroid terminology. Let M = ( E , I ) be a matroid. Definition. The rank of a subset X ⊆ E , denoted r ( X ), is the size of a maximal independent set contained in X . Thus r ( X ) � | X | for every X ⊆ E and X is independent if and only if | X | = r ( X ). Moreover, if X ⊆ Y then r ( X ) � r ( Y ). Let r ( M ) := r ( E ) be the rank of the matroid M . Orlando Lee – Unicamp Topics in Combinatorial Optimization
Rank Lemma. The rank function r is submodular, that is, r ( X ) + r ( Y ) � r ( X ∩ Y ) + r ( X ∪ Y ) , for every X , Y ⊆ E . Proof. Let I a maximal independent set of X ∩ Y . Thus | I | = r ( X ∩ Y ). Extend I to a maximal independent set J of X ∪ Y . Then | J | = r ( X ∪ Y ). The choice of I implies that J ∩ X ∩ Y = I and so | J ∩ X | + | J ∩ Y | = | I | + | J | . Since J ∩ X is an independent of X , we have r ( X ) � | J ∩ X | . Similarly, r ( Y ) � | J ∩ Y | . Then r ( X )+ r ( Y ) � | J ∩ X | + | J ∩ Y | = | I | + | J | = r ( X ∩ Y )+ r ( X ∪ Y ). Orlando Lee – Unicamp Topics in Combinatorial Optimization
Dimension Compare this result with the classic theorem from linear algebra relating the dimensions of two linear subspaces. Theorem. If V and W are linear (sub)spaces, then dim( V ) + dim( W ) = dim( V ∩ W ) + dim( V + W ) , where V + W := { v + w : v ∈ V , w ∈ W } . Also, if X and Y are subsets of columns of a matrix, then rank ( X ) + rank ( Y ) = rank ( X ∩ Y ) + rank ( X ∪ Y ) . and the rank function of a linear matroid is exactly the rank of matrices. Orlando Lee – Unicamp Topics in Combinatorial Optimization
Graphs Consider now a graph G = ( V , E ) and its associated graphic matroid M . Suppose for the moment that G is connected. What is the value of r ( M )? | V | − 1. Now let c ( G ) denote the number of components of G . What is the value of r ( M )? | V | − c ( G ). In general, for X ⊆ E we have r ( X ) = | V ( G [ X ]) | − c ( G [ X ]). Orlando Lee – Unicamp Topics in Combinatorial Optimization
Rank axioms Let us try to determine which properties a set-function r : 2 E �→ Z + must satisfy to be a rank function of a matroid. (R1) r ( ∅ ) = 0 (zero on the empty set), (R2) r ( X ) � r ( Y ) for X ⊆ Y ⊆ E (non-decreasing), (R3) r ( X ) � | X | (sub-cardinal), (R4) r ( X ) + r ( Y ) � r ( X ∩ Y ) + r ( X ∪ Y ) for every X , Y ⊆ E (submodular). We have seen that a rank function of a matroid satisfies all these conditions. Orlando Lee – Unicamp Topics in Combinatorial Optimization
Rank axioms Theorem. A set-function r : 2 E �→ Z + is the rank function of a matroid if and only if it satisfies (R1)(R2)(R3)(R4). Proof. We have shown that the conditions are necessary. Let us prove the sufficiency. Let I := { X ⊆ E : r ( X ) = | X |} . We will show that M := ( E , I ) is a matroid. First we prove the following: (R3’) r ( A + e ) � r ( A ) + 1 for every A ⊆ E and every e ∈ E − A . This follows from submodularity: r ( A ) + 1 � r ( A ) + r ( e ) � r ( A ∩ { e } ) + r ( A ∪ { e } ) � r ( A + e ). Orlando Lee – Unicamp Topics in Combinatorial Optimization
Rank axioms Lemma. Let A ⊆ E and e 1 , . . . , e k ∈ E − A . If r ( A + e i ) = r ( A ) for i = 1 , . . . , k , then r ( A ∪ { e 1 , . . . , e k } ) = r ( A ). (In other words, if the addition of elements cannot individually increase the rank of A , then neither can their simultaneous addition.) Proof. We use induction on k . If k = 1 the result is obvious. So suppose that k � 2 and the result holds for k − 1. Let A ′ := A ∪ { e 1 , . . . , e k − 1 } . By the induction hypothesis, we have r ( A ′ ) = r ( A ). From (R2) and (R4) we obtain r ( A ) + r ( A ) = r ( A + e k ) + r ( A ′ ) � r (( A + e k ) ∩ A ′ ) + r (( A + e k ) ∪ A ′ ) = r ( A ) + r ( A ∪ { e 1 , . . . , e k } ) � r ( A ) + r ( A ) , and the result follows. Orlando Lee – Unicamp Topics in Combinatorial Optimization
Rank axioms (R1) implies that (I1) holds. To prove that (I2) holds, let X ⊆ Y ∈ I with r ( Y ) = | Y | (that is, Y ∈ I ). By repeated applications of (R3’) we have r ( Y ) � r ( X ) + | Y − X | and hence r ( X ) � | X | . By (R3) it follows that r ( X ) = | X | and so X ∈ I . This proves (I2). Let us show that (I3’) holds, that is, every maximal subset of a set X which is in I has the same size. Let X ⊆ E . Take a maximal subset I of X which is in I (so r ( I ) = | I | ). We claim that | I | = r ( X ). Indeed, the maximality of I implies that I + e �∈ I for any e ∈ X − I . So r ( I ) � r ( I + e ) � | I | = r ( I ). So equality holds throughout and by the previous lemma we have | I | = r ( I ) = r ( X ). Thus every maximal subset of X and is in I has the same size r ( X ). So (I3’) holds. Orlando Lee – Unicamp Topics in Combinatorial Optimization
Other concepts A subset X ⊆ E is closed if r ( X + e ) > r ( X ) for every e ∈ E − X . A closed set is also called flat. The complement of a closed set is open. The ground set E is closed and ∅ is open. A closed set with rank r ( M ) − 1 is called hyperplane. The closure cl ( X ) of a subset X consists of the elements whose addition to X does not increase its rank. We say that cl ( X ) is spanned or generated by X . Equivalently, cl ( X ) is the unique largest superset of X that has the same rank of X . Or, cl ( X ) is the intersection of all closed sets containing X . Orlando Lee – Unicamp Topics in Combinatorial Optimization
Other concepts For a graph G = ( V , E ) let M := M ( G ) denote the graphic matroid associated with G . What are the closed sets of M ? What are the hyperplanes of M ? What is a closure of a subset X ⊆ E ? Orlando Lee – Unicamp Topics in Combinatorial Optimization
Exercises 1) Show that closed sets are closed under intersection. 2) Let I a maximal independent set of X . Prove that X is closed if and only if I + e is independent for every e ∈ E − X . Let G = ( V , E ) be a graph and let P a partition of V . The border of P is the set of edges that connect vertices in distinct members of P . 3) Consider a graphic matroid M := M ( G ) where G = ( V , E ). Show that a subset F ⊆ E is open if and only if it is the border of some partition of V . Orlando Lee – Unicamp Topics in Combinatorial Optimization
Maximal and minimal Let F a collection of sets. Let max F denote the collection of the maximal sets in F and let min F denote the collection of minimal sets in F . For a matroid M let I ( M ) , B ( M ) , C ( M ) and D ( M ) denote the collections of independent sets, bases, circuits and dependent sets of M . Then B ( M ) = max I ( M ) and C ( M ) = min D ( M ) − {∅} . Orlando Lee – Unicamp Topics in Combinatorial Optimization
Deletion Deletion. Let M := ( E , I ) be a matroid and S ⊆ E . The matroid M − S is the matroid with ground set E − S and independence set system: I ′ := { I − S : I ∈ I} . We say that M \ e is the matroid obtained from M by deleting S . (it is easy to see that I ′ satisfies the independence axioms.) When S = { e } we denote M \ e . The restriction of M to S is the matroid M \ ( E − S ). Orlando Lee – Unicamp Topics in Combinatorial Optimization
Deletion What does it mean to delete an element/set in a graphic matroid? And in a linear matroid? Consider the uniform matroid U n , k and let S be a S -element set of the ground set. Then � U n − s , n − s if n − k � s � n U n , k \ T ≡ if s < n − k . U n − s , k Orlando Lee – Unicamp Topics in Combinatorial Optimization
Deletion Theorem. Let M = ( E , I ) a matroid and let S ⊆ E . Then (a) I ( M \ S ) = { I − S : I ∈ I ( M ) } (definition), (b) B ( M \ S ) = max { B − S : B ∈ B ( M ) } , (c) C ( M \ S ) = { C ∈ E − S : C ∈ C ( M ) } , (d) r M \ S ( X ) = r M ( X ) for every X ⊆ E − S , (e) cl M \ S ( X ) = cl M ( X ) − S for every X ⊆ E − S . Exercise. Prove the theorem. Orlando Lee – Unicamp Topics in Combinatorial Optimization
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