The matching polytope has exponential extension complexity Thomas - PowerPoint PPT Presentation

b b b b b b b b b b b b b b Yannakakis’ Theorem Theorem (Yannakakis ’91) If S is the slack-matrix for P = { x ∈ R n | Ax ≤ b } , then xc( P ) = rk + ( S ). Factorization S = UV ⇒ extended formulation: ◮ Let P = { x ∈ R n | ∃ y ≥ 0 : Ax + Uy = b } Extended form. ⇒ factorization: ◮ Given an extension Q Q = { ( x, y ) | Bx + Cy ≤ d } ◮ For facet i : ( x j , y j ) A i x + 0 y ≤ b i u ( i ) := conic comb of i P ◮ For vertex x j : x j v ( j ) := d − Bx j − Cy j = slack of ( x j , y j ) � u ( i ) , v ( j ) � = u ( i ) T d − u ( i ) B x j − u ( i ) C y j = S ij � �� � � �� � � �� � = b i = A i = 0

Rectangle covering lower bound Observation rk + ( S ) ≥ rectangle-covering-number( S ).

Rectangle covering lower bound V 0 0 2 1 0 0 2 2 0 3 3 2 0 4 10 3 5 1 1 0 2 4 1 3 U 0 2 0 4 4 0 6 S 0 0 0 0 0 0 0 2 0 0 0 4 2 0 Observation rk + ( S ) ≥ rectangle-covering-number( S ).

Rectangle covering lower bound V 0 0 + + 0 0 + + 0 + + + 0 + + + + + + 0 + + + + U 0 + 0 + + 0 + S 0 0 0 0 0 0 0 + 0 0 + + 0 0 Observation rk + ( S ) ≥ rectangle-covering-number( S ).

Rectangle covering lower bound V 0 0 + + 0 0 + + 0 + + + 0 + + + + + + 0 + + + + U 0 + 0 + + 0 + S 0 0 0 0 0 0 0 + 0 0 + + 0 0 Observation rk + ( S ) ≥ rectangle-covering-number( S ).

Rectangle covering lower bound V 0 0 + + 0 0 + + 0 + + + 0 + + + + + + 0 + + + + U 0 + 0 + + 0 + S 0 0 0 0 0 0 0 + 0 0 + + 0 0 Observation rk + ( S ) ≥ rectangle-covering-number( S ).

Rectangle covering for matching ◮ Recall S U,M = | δ ( U ) ∩ M | − 1 Observation Rect-cov-num(matching polytope) ≤ O ( n 4 ). S cuts R e 1 ,e 2 matchings

Rectangle covering for matching ◮ Recall S U,M = | δ ( U ) ∩ M | − 1 Observation Rect-cov-num(matching polytope) ≤ O ( n 4 ). e 1 S e 2 cuts R e 1 ,e 2 matchings ◮ For e 1 , e 2 ∈ E :

Rectangle covering for matching ◮ Recall S U,M = | δ ( U ) ∩ M | − 1 Observation Rect-cov-num(matching polytope) ≤ O ( n 4 ). e 1 S U e 2 cuts R e 1 ,e 2 matchings ◮ For e 1 , e 2 ∈ E : take { U | e 1 , e 2 ∈ δ ( U ) }

Rectangle covering for matching ◮ Recall S U,M = | δ ( U ) ∩ M | − 1 Observation Rect-cov-num(matching polytope) ≤ O ( n 4 ). e 1 S U M e 2 cuts R e 1 ,e 2 matchings ◮ For e 1 , e 2 ∈ E : take { U | e 1 , e 2 ∈ δ ( U ) } ×{ M | e 1 , e 2 ∈ M }

Rectangle covering for matching ◮ Recall S U,M = | δ ( U ) ∩ M | − 1 Observation Rect-cov-num(matching polytope) ≤ O ( n 4 ). e 1 S U M e 2 cuts R e 1 ,e 2 . . . e k matchings ◮ For e 1 , e 2 ∈ E : take { U | e 1 , e 2 ∈ δ ( U ) } ×{ M | e 1 , e 2 ∈ M } � k � ◮ ( U, M ) with M ∩ δ ( U ) = { e 1 , . . . , e k } lies in rectangles 2

Rectangle covering for matching ◮ Recall S U,M = | δ ( U ) ∩ M | − 1 Observation Rect-cov-num(matching polytope) ≤ O ( n 4 ). e 1 S U M e 2 cuts R e 1 ,e 2 . . . e k matchings ◮ For e 1 , e 2 ∈ E : take { U | e 1 , e 2 ∈ δ ( U ) } ×{ M | e 1 , e 2 ∈ M } � k � ◮ ( U, M ) with M ∩ δ ( U ) = { e 1 , . . . , e k } lies in rectangles 2 0 1 1 � S ? R e 1 ,e 2 = 0 1 1 e 1 ,e 2 0 0 0

Rectangle covering for matching ◮ Recall S U,M = | δ ( U ) ∩ M | − 1 Observation Rect-cov-num(matching polytope) ≤ O ( n 4 ). e 1 S U M e 2 cuts R e 1 ,e 2 . . . e k matchings ◮ For e 1 , e 2 ∈ E : take { U | e 1 , e 2 ∈ δ ( U ) } ×{ M | e 1 , e 2 ∈ M } � k � ◮ ( U, M ) with M ∩ δ ( U ) = { e 1 , . . . , e k } lies in rectangles 2 ∼ k 2 S UM = k − 1 0 1 1 � S ? R e 1 ,e 2 = 0 1 1 e 1 ,e 2 0 0 0 | M ∩ δ ( U ) | = k

Rectangle covering for matching ◮ Recall S U,M = | δ ( U ) ∩ M | − 1 Observation Rect-cov-num(matching polytope) ≤ O ( n 4 ). e 1 S U M e 2 cuts R e 1 ,e 2 . . . e k matchings ◮ For e 1 , e 2 ∈ E : take { U | e 1 , e 2 ∈ δ ( U ) } ×{ M | e 1 , e 2 ∈ M } � k � ◮ ( U, M ) with M ∩ δ ( U ) = { e 1 , . . . , e k } lies in rectangles 2 Question Does every rectangle covering over-cover entries of large slack?

Rectangle covering for matching ◮ Recall S U,M = | δ ( U ) ∩ M | − 1 Observation Rect-cov-num(matching polytope) ≤ O ( n 4 ). e 1 S U M e 2 cuts R e 1 ,e 2 . . . e k matchings ◮ For e 1 , e 2 ∈ E : take { U | e 1 , e 2 ∈ δ ( U ) } ×{ M | e 1 , e 2 ∈ M } � k � ◮ ( U, M ) with M ∩ δ ( U ) = { e 1 , . . . , e k } lies in rectangles 2 Question Does every rectangle covering over-cover entries of large slack? YES!!

Hyperplane separation lower bound [Fiorini] ◮ Frobenius inner product: � W, S � := � � j W ij S ij i

b b b b b Hyperplane separation lower bound [Fiorini] ◮ Frobenius inner product: � W, S � := � � j W ij S ij i Lemma Pick W : � W, R � ≤ α ∀ rectangles R . R W 0 rectangles � W, R � ≤ α

b b b b b Hyperplane separation lower bound [Fiorini] ◮ Frobenius inner product: � W, S � := � � j W ij S ij i Lemma � W, S � Pick W : � W, R � ≤ α ∀ rectangles R . Then rk + ( S ) ≥ � S � ∞ · α R W 0 S rectangles � W, R � ≤ α

b b b b b Hyperplane separation lower bound [Fiorini] ◮ Frobenius inner product: � W, S � := � � j W ij S ij i Lemma � W, S � Pick W : � W, R � ≤ α ∀ rectangles R . Then rk + ( S ) ≥ � S � ∞ · α ◮ Proof: Write S = � r i =1 R i with rk + ( R i ) = 1. Then r r � � R i � � � W, S � = � R i � ∞ · W, ≤ α · � R i � ∞ ≤ α · r ·� S � ∞ . � R i � ∞ � �� � i =1 i =1 � �� � ≤� S � ∞ ≤ α R [0 , 1]-rank-1 matrices W 0 S rectangles � W, R � ≤ α

Applying the Hyperplane bound Lemma � W, S � Pick W : � W, R � ≤ α ∀ rectangles R . Then rk + ( S ) ≥ � S � ∞ · α

Applying the Hyperplane bound Lemma � W, S � Pick W : � W, R � ≤ α ∀ rectangles R . Then rk + ( S ) ≥ � S � ∞ · α ◮ Recall S UM = | δ ( U ) ∩ M | − 1

Applying the Hyperplane bound Lemma � W, S � Pick W : � W, R � ≤ α ∀ rectangles R . Then rk + ( S ) ≥ � S � ∞ · α ◮ Recall S UM = | δ ( U ) ∩ M | − 1 ◮ Abbreviate Q ℓ := { ( U, M ) : | δ ( U ) ∩ M | = ℓ }

Applying the Hyperplane bound Lemma � W, S � Pick W : � W, R � ≤ α ∀ rectangles R . Then rk + ( S ) ≥ � S � ∞ · α ◮ Recall S UM = | δ ( U ) ∩ M | − 1 ◮ Abbreviate Q ℓ := { ( U, M ) : | δ ( U ) ∩ M | = ℓ } ◮ Choose       W U,M =      0 otherwise .

Applying the Hyperplane bound Lemma � W, S � Pick W : � W, R � ≤ α ∀ rectangles R . Then rk + ( S ) ≥ � S � ∞ · α ◮ Recall S UM = | δ ( U ) ∩ M | − 1 ◮ Abbreviate Q ℓ := { ( U, M ) : | δ ( U ) ∩ M | = ℓ } ◮ Choose  − ∞ | δ ( U ) ∩ M | = 1      W U,M =      0 otherwise . ◮ Then � W, S � = 0

Applying the Hyperplane bound Lemma � W, S � Pick W : � W, R � ≤ α ∀ rectangles R . Then rk + ( S ) ≥ � S � ∞ · α ◮ Recall S UM = | δ ( U ) ∩ M | − 1 ◮ Abbreviate Q ℓ := { ( U, M ) : | δ ( U ) ∩ M | = ℓ } ◮ Choose  − ∞ | δ ( U ) ∩ M | = 1     1 | δ ( U ) ∩ M | = 3  | Q 3 | W U,M =      0 otherwise . ◮ Then � W, S � = 0 + 2

Applying the Hyperplane bound Lemma � W, S � Pick W : � W, R � ≤ α ∀ rectangles R . Then rk + ( S ) ≥ � S � ∞ · α ◮ Recall S UM = | δ ( U ) ∩ M | − 1 ◮ Abbreviate Q ℓ := { ( U, M ) : | δ ( U ) ∩ M | = ℓ } ◮ Choose  − ∞ | δ ( U ) ∩ M | = 1     1 | δ ( U ) ∩ M | = 3  | Q 3 | W U,M = 1 1 − k − 1 · | δ ( U ) ∩ M | = k   | Q k |    0 otherwise . ◮ Then � W, S � = 0 + 2 − 1 = 1

Applying the Hyperplane bound Lemma � W, S � Pick W : � W, R � ≤ α ∀ rectangles R . Then rk + ( S ) ≥ � S � ∞ · α ◮ Recall S UM = | δ ( U ) ∩ M | − 1 ◮ Abbreviate Q ℓ := { ( U, M ) : | δ ( U ) ∩ M | = ℓ } ◮ Choose  − ∞ | δ ( U ) ∩ M | = 1     1 | δ ( U ) ∩ M | = 3  | Q 3 | W U,M = 1 1 − k − 1 · | δ ( U ) ∩ M | = k   | Q k |    0 otherwise . ◮ Then � W, S � = 0 + 2 − 1 = 1 Lemma For k large, any rectangle R has � W, R � ≤ 2 − Ω( n ) .

Applying the Hyperplane bound (II) ◮ Uniform measure : µ ℓ ( R ) := | R ∩ Q ℓ | | Q ℓ |

Applying the Hyperplane bound (II) ◮ Uniform measure : µ ℓ ( R ) := | R ∩ Q ℓ | | Q ℓ | Main lemma ⇒ µ 3 ( R ) ≤ O ( 1 k 2 ) · µ k ( R ) + 2 − Ω( n ) µ 1 ( R ) = 0 = S cuts matchings

Applying the Hyperplane bound (II) ◮ Uniform measure : µ ℓ ( R ) := | R ∩ Q ℓ | | Q ℓ | Main lemma ⇒ µ 3 ( R ) ≤ O ( 1 k 2 ) · µ k ( R ) + 2 − Ω( n ) µ 1 ( R ) = 0 = S cuts R matchings

Applying the Hyperplane bound (II) ◮ Uniform measure : µ ℓ ( R ) := | R ∩ Q ℓ | | Q ℓ | Main lemma ⇒ µ 3 ( R ) ≤ O ( 1 k 2 ) · µ k ( R ) + 2 − Ω( n ) µ 1 ( R ) = 0 = S cuts R matchings

Applying the Hyperplane bound (II) ◮ Uniform measure : µ ℓ ( R ) := | R ∩ Q ℓ | | Q ℓ | Main lemma ⇒ µ 3 ( R ) ≤ O ( 1 k 2 ) · µ k ( R ) + 2 − Ω( n ) µ 1 ( R ) = 0 = S cuts R matchings

Applying the Hyperplane bound (II) ◮ Uniform measure : µ ℓ ( R ) := | R ∩ Q ℓ | | Q ℓ | Main lemma ⇒ µ 3 ( R ) ≤ O ( 1 k 2 ) · µ k ( R ) + 2 − Ω( n ) µ 1 ( R ) = 0 = S cuts R matchings

Applying the Hyperplane bound (II) ◮ Uniform measure : µ ℓ ( R ) := | R ∩ Q ℓ | | Q ℓ | Main lemma ⇒ µ 3 ( R ) ≤ O ( 1 k 2 ) · µ k ( R ) + 2 − Ω( n ) µ 1 ( R ) = 0 = S cuts R matchings ◮ Technique: Partition scheme [Razborov ’91]

Applying the Hyperplane bound (II) ◮ Uniform measure : µ ℓ ( R ) := | R ∩ Q ℓ | | Q ℓ | Main lemma ⇒ µ 3 ( R ) ≤ O ( 1 k 2 ) · µ k ( R ) + 2 − Ω( n ) µ 1 ( R ) = 0 = S cuts T R matchings ◮ Technique: Partition scheme [Razborov ’91]

Partitions S cuts ◮ Partition T = ( A, C, D, B ) T R matchings

Partitions S cuts ◮ Partition T = ( A, C, D, B ) T R A matchings

Partitions S cuts ◮ Partition T = ( A, C, D, B ) T R A B matchings

Partitions S cuts ◮ Partition T = ( A, C, D, B ) T R A C B matchings k

Partitions S cuts ◮ Partition T = ( A, C, D, B ) T R A C D B matchings k k

Partitions S cuts ◮ Partition T = ( A, C, D, B ) T R A C D B matchings A 1 . . . k − 3 A m nodes k k

Partitions S cuts ◮ Partition T = ( A, C, D, B ) T R A C D B matchings . . . A 1 B 1 B m . . . k − 3 A m nodes 2( k − 3) k k nodes

Partitions S cuts ◮ Partition T = ( A, C, D, B ) T R ◮ Edges E ( T ) A C D B matchings . . . A 1 B 1 B m . . . k − 3 A m nodes 2( k − 3) k k nodes

Partitions S cuts ◮ Partition T = ( A, C, D, B ) T R ◮ Edges E ( T ) A C D B matchings . . . A 1 B 1 B m . . . k − 3 A m nodes 2( k − 3) k k nodes

Partitions S cuts ◮ Partition T = ( A, C, D, B ) T R ◮ Edges E ( T ) A C D B matchings U . . . A 1 B 1 B m . . . k − 3 A m nodes 2( k − 3) k k nodes

S Rewriting µ 3 ( R ) cuts T R matchings Randomly generate ( U, M ) ∼ Q 3 : µ 3 ( R ) =

S Rewriting µ 3 ( R ) cuts A C D B T R . . . A 1 B 1 B m matchings . . . A m Randomly generate ( U, M ) ∼ Q 3 : 1. Choose T � � µ 3 ( R ) = E T

S Rewriting µ 3 ( R ) cuts A C D B T R H . . . A 1 B 1 B m matchings . . . A m Randomly generate ( U, M ) ∼ Q 3 : 1. Choose T 2. Choose 3 edges H ⊆ C × D � � �� µ 3 ( R ) = E E T | H | =3

S Rewriting µ 3 ( R ) cuts A C D B T R H . . . A 1 B 1 B m matchings . . . A m Randomly generate ( U, M ) ∼ Q 3 : 1. Choose T 2. Choose 3 edges H ⊆ C × D 3. Choose M ⊇ H (not cutting any other edge in C × D ) � � �� µ 3 ( R ) = E E T | H | =3

S Rewriting µ 3 ( R ) cuts A C D B T R U H . . . A 1 B 1 B m matchings . . . A m Randomly generate ( U, M ) ∼ Q 3 : 1. Choose T 2. Choose 3 edges H ⊆ C × D 3. Choose M ⊇ H (not cutting any other edge in C × D ) 4. Choose U cutting H (not cutting any A i ) � � �� µ 3 ( R ) = E Pr[( U, M ) ∈ R | T, H ] E T | H | =3

S Rewriting µ 3 ( R ) cuts A C D B T R U H . . . A 1 B 1 B m matchings . . . A m Randomly generate ( U, M ) ∼ Q 3 : 1. Choose T 2. Choose 3 edges H ⊆ C × D 3. Choose M ⊇ H (not cutting any other edge in C × D ) 4. Choose U cutting H (not cutting any A i ) � � �� µ 3 ( R ) = E Pr[ U ∈ R | T, H ] · Pr[ M ∈ R | T, H ] E T | H | =3

S Rewriting µ k ( R ) cuts T R matchings Randomly generate ( U, M ) ∼ Q k : µ k ( R ) =

S Rewriting µ k ( R ) cuts A C D B T R . . . A 1 B 1 B m matchings . . . A m Randomly generate ( U, M ) ∼ Q k : 1. Choose T � � µ k ( R ) = E T

S Rewriting µ k ( R ) cuts A C D B T R F . . . A 1 B 1 B m matchings . . . A m Randomly generate ( U, M ) ∼ Q k : 1. Choose T 2. Choose k edges F ⊆ C × D � � �� µ k ( R ) = E E T | F | = k

S Rewriting µ k ( R ) cuts A C D B T R F . . . A 1 B 1 B m matchings . . . A m Randomly generate ( U, M ) ∼ Q k : 1. Choose T 2. Choose k edges F ⊆ C × D 3. Choose M ⊇ F � � �� µ k ( R ) = E Pr[ M ∈ R | T, H ] E T | F | = k

S Rewriting µ k ( R ) cuts A C D B T R U F . . . A 1 B 1 B m matchings . . . A m Randomly generate ( U, M ) ∼ Q k : 1. Choose T 2. Choose k edges F ⊆ C × D 3. Choose M ⊇ F 4. Choose U ⊇ C (not cutting any A i ) � � �� µ k ( R ) = E Pr[ M ∈ R | T, H ] · Pr[ U ∈ R | T, H ] E T | F | = k

Pseudorandom-behaviour of large sets ◮ Consider vectors X ⊆ [ q ] n . 1 1 1 1 1 1 1 2 2 2 2 2 2 2 .. .. .. .. .. .. .. q q q q q q q . . . n 1 2

Pseudorandom-behaviour of large sets ◮ Consider vectors X ⊆ [ q ] n . ◮ Draw x ∼ X . 1 1 1 1 1 1 1 2 2 2 2 2 2 2 .. .. .. .. .. .. .. q q q q q q q . . . n 1 2

Pseudorandom-behaviour of large sets ◮ Consider vectors X ⊆ [ q ] n . ◮ Draw x ∼ X . 1 1 1 1 1 1 1 2 2 2 2 2 2 2 .. .. .. .. .. .. .. q q q q q q q . . . n 1 2 i

Pseudorandom-behaviour of large sets ◮ Consider vectors X ⊆ [ q ] n . ◮ Draw x ∼ X . 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 .. .. .. .. .. .. .. .. .. .. q q q q q q q q . . . n 1 2 i

Pseudorandom-behaviour of large sets ◮ Consider vectors X ⊆ [ q ] n . ◮ Draw x ∼ X . 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 .. .. .. .. .. .. .. .. .. q q q q q q q q q . . . n 1 2 i

Pseudorandom-behaviour of large sets ◮ Consider vectors X ⊆ [ q ] n . ◮ Draw x ∼ X . Lemma | X | large ⇒ for most indices x i is approx. uniform 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 .. .. .. .. .. .. .. .. .. q q q q q q q q q . . . n 1 2 i

Pseudorandom-behaviour of large sets ◮ Consider vectors X ⊆ [ q ] n . ◮ Draw x ∼ X . Lemma εn biased indices ⇒ | X | q n ≤ 2 − Ω( n ) . 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 .. .. .. .. .. .. .. .. .. q q q q q q q q q . . . n 1 2 i

Pseudorandom-behaviour of large sets ◮ Consider vectors X ⊆ [ q ] n . ◮ Draw x ∼ X . Lemma εn biased indices ⇒ | X | q n ≤ 2 − Ω( n ) . log 2 ( | X | ) = H ( x ) 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 .. .. .. .. .. .. .. .. .. q q q q q q q q q . . . n 1 2 i

Pseudorandom-behaviour of large sets ◮ Consider vectors X ⊆ [ q ] n . ◮ Draw x ∼ X . Lemma εn biased indices ⇒ | X | q n ≤ 2 − Ω( n ) . n � log 2 ( | X | ) = H ( x ) ≤ H ( x i ) i =1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 .. .. .. .. .. .. .. .. .. q q q q q q q q q . . . n 1 2 i