Greedy Algorithm and Matroid Intersections by Yan Alves Radtke July - PowerPoint PPT Presentation

Maximum weight algorithm for matroid intersections Input: M 1 and M 2 , an extreme common independent set I and a weight function w Output: An extreme common independent set J with | J | = | I | + 1 if any exists, else I Construct D M 1 , M 2 ( I ); Define X i := { x ∈ S \ I | I ∪ { x } ∈ I i } ; if X 1 -X 2 Path exists in D M 1 , M 2 ( I ) then P:=minimal weight X 1 - X 2 Path with minimal number of arcs; return I △ VP else return I end by Yan Alves Radtke Greedy Algorithm and Matroid Intersections

Matchings I S \ I by Yan Alves Radtke Greedy Algorithm and Matroid Intersections

Perfect Matchings I S \ I by Yan Alves Radtke Greedy Algorithm and Matroid Intersections

Not Perfect Matchings I S \ I by Yan Alves Radtke Greedy Algorithm and Matroid Intersections

Properties of perfect matchings If J ∈ I 1 and | J | = | I | , there exists a series of swaps that transform J into I by Yan Alves Radtke Greedy Algorithm and Matroid Intersections

Properties of perfect matchings If J ∈ I 1 and | J | = | I | , there exists a series of swaps that transform J into I = ⇒ D M 1 , M 2 ( I ) has a perfect matching with only downward edges on I △ J by Yan Alves Radtke Greedy Algorithm and Matroid Intersections

Properties of perfect matchings If J ∈ I 1 and | J | = | I | , there exists a series of swaps that transform J into I = ⇒ D M 1 , M 2 ( I ) has a perfect matching with only downward edges on I △ J If | J | = | I | and D M 1 , M 2 ( I ) has a unique perfect matching on I △ J with only downward edges by Yan Alves Radtke Greedy Algorithm and Matroid Intersections

Properties of perfect matchings If J ∈ I 1 and | J | = | I | , there exists a series of swaps that transform J into I = ⇒ D M 1 , M 2 ( I ) has a perfect matching with only downward edges on I △ J If | J | = | I | and D M 1 , M 2 ( I ) has a unique perfect matching on I △ J with only downward edges = ⇒ J ∈ I 1 by Yan Alves Radtke Greedy Algorithm and Matroid Intersections

Lemma Let C ∋ α be a circuit s.t. I △ VC is not a common independent set α by Yan Alves Radtke Greedy Algorithm and Matroid Intersections

Lemma Let C ∋ α be a circuit s.t. I △ VC is not a common independent set α Then there exists: Negative length circuit C ′ with VC ′ � VC or C ′ ∋ α s.t. ℓ ( C ′ ) ≤ ℓ ( C ) with VC ′ � VC by Yan Alves Radtke Greedy Algorithm and Matroid Intersections

Proof of Lemma α Matching by Yan Alves Radtke Greedy Algorithm and Matroid Intersections

Proof of Lemma α Matching Since not independent: another matching by Yan Alves Radtke Greedy Algorithm and Matroid Intersections

Proof of Lemma α Matching Since not independent: another matching by Yan Alves Radtke Greedy Algorithm and Matroid Intersections

Proof of Lemma Eulerian Graph = ⇒ decomposition into circuits C 1 , C 2 ..., C j by Yan Alves Radtke Greedy Algorithm and Matroid Intersections

Proof of Lemma Eulerian Graph = ⇒ decomposition into circuits C 1 , C 2 ..., C j s.t. � ℓ ( C i ) = 2 ℓ ( C ) i =1 ,..., j by Yan Alves Radtke Greedy Algorithm and Matroid Intersections

Proof of Lemma Eulerian Graph = ⇒ decomposition into circuits C 1 , C 2 ..., C j s.t. � ℓ ( C i ) = 2 ℓ ( C ) i =1 ,..., j if there is no negative weight circuit, it follows, if t ∈ C 1 , C 2 : by Yan Alves Radtke Greedy Algorithm and Matroid Intersections

Proof of Lemma Eulerian Graph = ⇒ decomposition into circuits C 1 , C 2 ..., C j s.t. � ℓ ( C i ) = 2 ℓ ( C ) i =1 ,..., j if there is no negative weight circuit, it follows, if t ∈ C 1 , C 2 : ℓ ( C 1 ) + ℓ ( C 2 ) ≤ � ℓ ( C i ) = 2 ℓ ( C ) i =1 ,..., j by Yan Alves Radtke Greedy Algorithm and Matroid Intersections

Proof of Lemma Eulerian Graph = ⇒ decomposition into circuits C 1 , C 2 ..., C j s.t. � ℓ ( C i ) = 2 ℓ ( C ) i =1 ,..., j if there is no negative weight circuit, it follows, if t ∈ C 1 , C 2 : ℓ ( C 1 ) + ℓ ( C 2 ) ≤ � ℓ ( C i ) = 2 ℓ ( C ) i =1 ,..., j = ⇒ ℓ ( C 1 ) ≤ ℓ ( C ) or ℓ ( C 2 ) ≤ ℓ ( C ) by Yan Alves Radtke Greedy Algorithm and Matroid Intersections

Extreme set - negative circuit theorem Statement: D M 1 , M 2 ( I ) has no negative length circuit ⇔ I is an extreme set by Yan Alves Radtke Greedy Algorithm and Matroid Intersections

Proof ⇒ Let J be a common independet set with | J | = | I | I \ J J \ I by Yan Alves Radtke Greedy Algorithm and Matroid Intersections

Proof ⇒ � w ( J ) = w ( I ) − ℓ ( I △ J ) = w ( I ) − ℓ ( C i ) ≤ w ( I ) by Yan Alves Radtke Greedy Algorithm and Matroid Intersections

Proof ⇒ � w ( J ) = w ( I ) − ℓ ( I △ J ) = w ( I ) − ℓ ( C i ) ≤ w ( I ) = ⇒ w ( J ) ≤ w ( I ) by Yan Alves Radtke Greedy Algorithm and Matroid Intersections

Proof ⇒ � w ( J ) = w ( I ) − ℓ ( I △ J ) = w ( I ) − ℓ ( C i ) ≤ w ( I ) = ⇒ w ( J ) ≤ w ( I ) = ⇒ I is an extreme set by Yan Alves Radtke Greedy Algorithm and Matroid Intersections

Proof ⇐ Let C negative length circuit minimal nodes and I extreme set by Yan Alves Radtke Greedy Algorithm and Matroid Intersections

Proof ⇐ Let C negative length circuit minimal nodes and I extreme set Then w ( I △ VC ) = w ( I ) − ℓ ( VC ) > w ( I ) by Yan Alves Radtke Greedy Algorithm and Matroid Intersections

Proof ⇐ Let C negative length circuit minimal nodes and I extreme set Then w ( I △ VC ) = w ( I ) − ℓ ( VC ) > w ( I ) Then I △ VC is not a commmon independent set by Yan Alves Radtke Greedy Algorithm and Matroid Intersections

Reminder of Lemma Let C ∋ t be a circuit s.t. I △ VC is not a common independent set α Then there exists: Negative length circuit C ′ with VC ′ � VC or C ′ ∋ α s.t. ℓ ( C ′ ) ≤ ℓ ( C ) with VC ′ � VC by Yan Alves Radtke Greedy Algorithm and Matroid Intersections

Reminder of Lemma Let C ∋ t be a circuit s.t. I △ VC is not a common independent set α Since ℓ ( C ) < 0 this implies: Negative length circuit C ′ with VC ′ � VC by Yan Alves Radtke Greedy Algorithm and Matroid Intersections

Proof ⇐ Let C negative length circuit minimal nodes and I extreme set Then w ( I △ VC ) = w ( I ) − ℓ ( VC ) > w ( I ) Then I △ VC is not a commmon independent set By Lemma C ′ is a negative length circuit with less nodes � by Yan Alves Radtke Greedy Algorithm and Matroid Intersections

Maximum weight algorithm Input: M 1 and M 2 , an extreme common independent set I and a weight function w Output: An extreme common independent set J with | J | = | I | + 1 if any exists, else I Construct D M 1 , M 2 ( I ); Define X i := { x ∈ S \ I | I ∪ { x } ∈ I i } ; if X 1 -X 2 Path exists in D M 1 , M 2 ( I ) then P:=minimal weight X 1 - X 2 Path with minimal number of arcs; return I △ VP else return I end by Yan Alves Radtke Greedy Algorithm and Matroid Intersections

Auxiliary Matroid Def. M ′ i := ( S + t , { U ⊆ S + t | U − t ∈ I i } ) by Yan Alves Radtke Greedy Algorithm and Matroid Intersections

Auxiliary Matroid Def. M ′ i := ( S + t , { U ⊆ S + t | U − t ∈ I i } ) Claim D M ′ 2 ( I + t )[ S ] = D M 1 , M 2 ( I ) 1 , M ′ by Yan Alves Radtke Greedy Algorithm and Matroid Intersections

Auxiliary Matroid Def. M ′ i := ( S + t , { U ⊆ S + t | U − t ∈ I i } ) Claim D M ′ 2 ( I + t )[ S ] = D M 1 , M 2 ( I ) 1 , M ′ Proof I + t − x + y ∈ I ′ i ⇔ I − x + y ∈ I i by Yan Alves Radtke Greedy Algorithm and Matroid Intersections

Auxiliary Matroid Def. M ′ i := ( S + t , { U ⊆ S + t | U − t ∈ I i } ) Claim D M ′ 2 ( I + t )[ S ] = D M 1 , M 2 ( I ) 1 , M ′ Proof I + t − x + y ∈ I ′ i ⇔ I − x + y ∈ I i Claim N ( t ) = X 1 ∪ X 2 by Yan Alves Radtke Greedy Algorithm and Matroid Intersections

Auxiliary Matroid Def. M ′ i := ( S + t , { U ⊆ S + t | U − t ∈ I i } ) Claim D M ′ 2 ( I + t )[ S ] = D M 1 , M 2 ( I ) 1 , M ′ Proof I + t − x + y ∈ I ′ i ⇔ I − x + y ∈ I i Claim N ( t ) = X 1 ∪ X 2 Proof I + t − t + x ∈ I ′ i ⇔ I + x ∈ I ′ i ⇔ I + x ∈ I i ⇔ x ∈ X i by Yan Alves Radtke Greedy Algorithm and Matroid Intersections

Proof of extremity of I △ VP t x ′ x ′ 1 2 x 1 x 2 I I x ′ x ′ 1 2 by Yan Alves Radtke Greedy Algorithm and Matroid Intersections

Proof of extremity of I △ VP w ( t ) := − ℓ ( P ) t x ′ x ′ 1 2 x 1 x 2 I I x ′ x ′ 1 2 by Yan Alves Radtke Greedy Algorithm and Matroid Intersections

Proof of extremity of I △ VP w ( t ) := − ℓ ( P ) t x ′ x ′ 1 2 x 1 x 2 I I ℓ ( P ′ ) ≥ ℓ ( P ) x ′ x ′ 1 2 by Yan Alves Radtke Greedy Algorithm and Matroid Intersections

Proof of extremity of I △ VP w ( I + t ) = w ( I ) + w ( t ) = w ( I ) − ℓ ( P ) = w ( I △ P ) = w ( J ) by Yan Alves Radtke Greedy Algorithm and Matroid Intersections

Proof of extremity of I △ VP w ( I + t ) = w ( I ) + w ( t ) = w ( I ) − ℓ ( P ) = w ( I △ P ) = w ( J ) A relaxation of our original problem has a maximum weight of w ( J ) by Yan Alves Radtke Greedy Algorithm and Matroid Intersections

Proof of extremity of I △ VP w ( I + t ) = w ( I ) + w ( t ) = w ( I ) − ℓ ( P ) = w ( I △ P ) = w ( J ) A relaxation of our original problem has a maximum weight of w ( J ) J common independent = ⇒ J is extreme common independent by Yan Alves Radtke Greedy Algorithm and Matroid Intersections

Proof of independency of I △ VP w ( t ) := − ℓ ( P ) t x ′ x ′ 1 2 x 1 x 2 I I x ′ x ′ 1 2 by Yan Alves Radtke Greedy Algorithm and Matroid Intersections

Reminder of Lemma Let C ∋ t be a circuit s.t. I △ VC is not a common independent set α Then there exists: Negative length circuit C ′ with VC ′ � VC or C ′ ∋ t s.t. ℓ ( C ′ ) ≤ ℓ ( C ) with VC ′ � VC by Yan Alves Radtke Greedy Algorithm and Matroid Intersections

Proof of independence of I △ VP w ( t ) := − ℓ ( P ) t x ′ x ′ 1 2 x 1 x 2 I I | P ′ | > = | P | or ℓ ( P ′ ) > ℓ ( P ) x ′ x ′ 1 2 by Yan Alves Radtke Greedy Algorithm and Matroid Intersections

Reminder of Lemma Let C ∋ α be a circuit s.t. I △ VC is not a common independent set α Then there exists: Negative length circuit C ′ with VC ′ � VC or C ′ ∋ t s.t. ℓ ( C ′ ) ≤ ℓ ( C ) with VC ′ � VC by Yan Alves Radtke Greedy Algorithm and Matroid Intersections

Reminder of Lemma Let C ∋ α be a circuit s.t. I △ VC is not a common independent set α Then there exists: Negative length circuit C ′ with VC ′ � VC or C ′ ∋ t s.t. ℓ ( C ′ ) ≤ ℓ ( C ) with VC ′ � VC by Yan Alves Radtke Greedy Algorithm and Matroid Intersections

Proof of independence ( I + t △ VP + t ) = I △ VP is a common independent set by Yan Alves Radtke Greedy Algorithm and Matroid Intersections

Proof of independence ( I + t △ VP + t ) = I △ VP is a common independent set = ⇒ J = I △ VP is an extreme common independent set by Yan Alves Radtke Greedy Algorithm and Matroid Intersections

Maximum weight algorithm Input: M 1 and M 2 , an extreme common independent set I and a weight function w Output: An extreme common independent set J with | J | = | I | + 1 if any exists, else I Construct D M 1 , M 2 ( I ); Define X i := { x ∈ S \ I | I ∪ { x } ∈ I i } ; if X 1 -X 2 Path exists in D M 1 , M 2 ( I ) then P:=minimal weight X 1 - X 2 Path with minimal number of arcs; return I △ VP else return I end by Yan Alves Radtke Greedy Algorithm and Matroid Intersections

Case 2 X 1

Case 2 X 1 X 2

Case 2 X 1 X 2