The Rectangle Covering number of Random Boolean Matrices Mozhgan - PowerPoint PPT Presentation

The Rectangle Covering number of Random Boolean Matrices Mozhgan Pourmoradnasseri University of Tartu Estonia Dirk Oliver Theis University of Tartu Estonia Estonian Computer Science Theory Day October 4, 2015

What this talk is about Complexity Theory Combinatorics Combinatorial Matrix Theory Nondeterministic Rectangle Biclique Boolean Rank Communication Covering Covering of a Boolean Complexity of a Number of a Number of a matrix Boolean function f 01 matrix M bipartite graph H smallest smallest . . . number of number of . . . 1-rectangles of blicliques needed needed to cover all to cover all 1-entries edges in M of H Def.: 1-Rectangle in M : K × L w/ M k,ℓ = 1 ∀ ( k, ℓ ) ∈ K × L

What this talk is about Background ◮ Rectangle Covering Number, rc( M ) , of a 01-matrix M This talk:

What this talk is about Background ◮ Rectangle Covering Number, rc( M ) , of a 01-matrix M rc( M ) Random 01-matrices M n,p ◮ (entries Bernoulli w/ parameter p ) � � chromatic number of special type of “Lov´ asz-Saks random graphs rectangle graph of M ” G n,p ⊠ G ⊠ ( M ) This talk:

What this talk is about Background ◮ Rectangle Covering Number, rc( M ) , of a 01-matrix M rc( M ) Random 01-matrices M n,p ◮ (entries Bernoulli w/ parameter p ) � � chromatic number of special type of “Lov´ asz-Saks random graphs rectangle graph of M ” G n,p ⊠ G ⊠ ( M ) This talk:

What this talk is about Background ◮ Rectangle Covering Number, rc( M ) , of a 01-matrix M rc( M ) Random 01-matrices M n,p ◮ (entries Bernoulli w/ parameter p ) � � chromatic number of special type of “Lov´ asz-Saks random graphs rectangle graph of M ” G n,p ⊠ G ⊠ ( M ) This talk: ◮ Bound rc( M n,p ) = χ ( G n,p ⊠ )

What this talk is about Background ◮ Rectangle Covering Number, rc( M ) , of a 01-matrix M rc( M ) Random 01-matrices M n,p ◮ (entries Bernoulli w/ parameter p ) � � chromatic number of special type of “Lov´ asz-Saks random graphs rectangle graph of M ” G n,p ⊠ G ⊠ ( M ) This talk: ◮ Bound rc( M n,p ) = χ ( G n,p ⊠ ) ◮ Bound other parameters related to χ , e.g., clique number, independence number.

Outline Lov´ asz-Saks Rectangle Graph G ⊠ ( M ) Basics on the random graphs G n,p ⊠ The clique number ( = fooling set size) The independence number ( = largest 1-rectangle) The chromatic number ( = rc , ˆ = NdCC)

Lov´ asz-Saks Rectangle Graph G ⊠ ( M ) Basics on the random graphs G n,p ⊠ The clique number ( = fooling set size) The independence number ( = largest 1-rectangle) The chromatic number ( = rc , ˆ = NdCC)

Lov´ asz-Saks Rectangle Graph G ⊠ ( M ) − → 01 matrix M Graph G ⊠ ( M )  ∗ ∗ ∗ ∗ ∗  ∗ ∗ ∗ ∗ 1     ∗ ∗ ∗ ∗ ∗     ∗ ∗ ∗ ∗ 1   ∗ ∗ ∗ ∗ ∗ adjacent

Lov´ asz-Saks Rectangle Graph G ⊠ ( M ) 01 matrix M Graph G ⊠ ( M ) − → Vertices of G ⊠ ( M ) : 1-entries  ∗ ∗ ∗ ∗ ∗  ∗ ∗ ∗ ∗ 1     ∗ ∗ ∗ ∗ ∗     ∗ ∗ ∗ ∗ � � 1 V = ( k, ℓ ) | M k,ℓ = 1   ∗ ∗ ∗ ∗ ∗ adjacent

Lov´ asz-Saks Rectangle Graph G ⊠ ( M ) 01 matrix M Graph G ⊠ ( M ) − → Vertices of G ⊠ ( M ) : 1-entries  ∗ ∗ ∗ ∗ ∗  ∗ ∗ ∗ ∗ 1     ∗ ∗ ∗ ∗ ∗     ∗ ∗ ∗ ∗ � � 1 V = ( k, ℓ ) | M k,ℓ = 1   ∗ ∗ ∗ ∗ ∗ adjacent Edges of G ⊠ ( M ) : “Two 1-entries (= vertices) are adjacent, if they span a 2 × 2 rectangle containing a 0.” � � { ( k, ℓ ) , ( k ′ , ℓ ′ ) } | M k,ℓ M k ′ ,ℓ ′ = 1 M k ′ ,ℓ M k,ℓ ′ = 0 E = &

Lov´ asz-Saks Rectangle Graph G ⊠ ( M ) 01 matrix M Graph G ⊠ ( M ) − → Vertices of G ⊠ ( M ) : 1-entries  ∗ ∗ ∗ ∗ ∗  ∗ ∗ ∗ 1 ??     ∗ ∗ ∗ ∗ ∗     ∗ ∗ ∗ � � ?? 1 V = ( k, ℓ ) | M k,ℓ = 1   ∗ ∗ ∗ ∗ ∗ adjacent Edges of G ⊠ ( M ) : “Two 1-entries (= vertices) are adjacent, if they span a 2 × 2 rectangle containing a 0.” � � { ( k, ℓ ) , ( k ′ , ℓ ′ ) } | M k,ℓ M k ′ ,ℓ ′ = 1 M k ′ ,ℓ M k,ℓ ′ = 0 E = &

Lov´ asz-Saks Rectangle Graph G ⊠ ( M ) 01 matrix M Graph G ⊠ ( M ) − → Vertices of G ⊠ ( M ) : 1-entries  ∗ ∗ ∗ ∗ ∗  ∗ ∗ ∗ 1 1     ∗ ∗ ∗ ∗ ∗     ∗ ∗ ∗ � � 1 1 V = ( k, ℓ ) | M k,ℓ = 1   ∗ ∗ ∗ ∗ ∗ not adjacent Edges of G ⊠ ( M ) : “Two 1-entries (= vertices) are adjacent, if they span a 2 × 2 rectangle containing a 0.” � � { ( k, ℓ ) , ( k ′ , ℓ ′ ) } | M k,ℓ M k ′ ,ℓ ′ = 1 M k ′ ,ℓ M k,ℓ ′ = 0 E = &

Lov´ asz-Saks Rectangle Graph G ⊠ ( M ) 01 matrix M Graph G ⊠ ( M ) − → Vertices of G ⊠ ( M ) : 1-entries  ∗ ∗ ∗ ∗ ∗  ∗ ∗ ∗ 1 0     ∗ ∗ ∗ ∗ ∗     ∗ ∗ ∗ � � 1 1 V = ( k, ℓ ) | M k,ℓ = 1   ∗ ∗ ∗ ∗ ∗ adjacent Edges of G ⊠ ( M ) : “Two 1-entries (= vertices) are adjacent, if they span a 2 × 2 rectangle containing a 0.” � � { ( k, ℓ ) , ( k ′ , ℓ ′ ) } | M k,ℓ M k ′ ,ℓ ′ = 1 M k ′ ,ℓ M k,ℓ ′ = 0 E = &

Lov´ asz-Saks Rectangle Graph G ⊠ ( M ) 01 matrix M Graph G ⊠ ( M ) − → Vertices of G ⊠ ( M ) : 1-entries  ∗ ∗ ∗ ∗ ∗  ∗ ∗ ∗ 1 1     ∗ ∗ ∗ ∗ ∗     ∗ ∗ ∗ � � 0 1 V = ( k, ℓ ) | M k,ℓ = 1   ∗ ∗ ∗ ∗ ∗ adjacent Edges of G ⊠ ( M ) : “Two 1-entries (= vertices) are adjacent, if they span a 2 × 2 rectangle containing a 0.” � � { ( k, ℓ ) , ( k ′ , ℓ ′ ) } | M k,ℓ M k ′ ,ℓ ′ = 1 M k ′ ,ℓ M k,ℓ ′ = 0 E = &

Lov´ asz-Saks Rectangle Graph G ⊠ ( M ) 01 matrix M Graph G ⊠ ( M ) − → Vertices of G ⊠ ( M ) : 1-entries  ∗ ∗ ∗ ∗ ∗  ∗ ∗ ∗ 1 0     ∗ ∗ ∗ ∗ ∗     ∗ ∗ ∗ � � 0 1 V = ( k, ℓ ) | M k,ℓ = 1   ∗ ∗ ∗ ∗ ∗ adjacent Edges of G ⊠ ( M ) : “Two 1-entries (= vertices) are adjacent, if they span a 2 × 2 rectangle containing a 0.” � � { ( k, ℓ ) , ( k ′ , ℓ ′ ) } | M k,ℓ M k ′ ,ℓ ′ = 1 M k ′ ,ℓ M k,ℓ ′ = 0 E = &

Rectangle Graph — Example Definition of rectangle graph G ⊠ ( M ) : � � � V := ( k, ℓ ) � M k,ℓ = 1 = set of 1-entries of M � E : ( k, ℓ ) ∼ ( k ′ , ℓ ′ ) iff M k,ℓ ′ M k ′ ,ℓ = 0 Example  1 0 0      0 1 1         1 1 0       1 1 1

Rectangle Graph — Example Definition of rectangle graph G ⊠ ( M ) : � � � V := ( k, ℓ ) � M k,ℓ = 1 = set of 1-entries of M � E : ( k, ℓ ) ∼ ( k ′ , ℓ ′ ) iff M k,ℓ ′ M k ′ ,ℓ = 0 Example  1 0 0      0 1 1         1 1 0       1 1 1

Rectangle Graph — Example Definition of rectangle graph G ⊠ ( M ) : � � � V := ( k, ℓ ) � M k,ℓ = 1 = set of 1-entries of M � E : ( k, ℓ ) ∼ ( k ′ , ℓ ′ ) iff M k,ℓ ′ M k ′ ,ℓ = 0 Example  1 0 0      0 1 1         1 1 0       1 1 1

Relationship with graph coloring Lov´ asz-Saks (1993): rc( M ) = χ ( G ⊠ ( M )) Because: Inclusion-wise maximal independent sets in G ⊠ ( M ) � � inclusion-wise maximal 1-rectangles in M

Relationship with graph coloring Lov´ asz-Saks (1993): rc( M ) = χ ( G ⊠ ( M )) Because: Inclusion-wise maximal independent sets in G ⊠ ( M ) � � inclusion-wise maximal 1-rectangles in M Construction “goes in both directions”: ◮ ∀ G : G induced subgraph of G ⊠ ( 1 − Adj( G ))  1 . . . 1  1 = . . ...   . .  all-1 matrix of appropriate dimensions.   . .  1 . . . 1

The Log-Rank conjecture Log-Rank Conjecture, Communication Complexity version Deterministic CC ( M ) ≤ polylog rk( M ) Log-Rank Conjecture, Graph Theory version χ ( G ) ≤ 2 polylog rk(Adj( G )) (The two are equivalent though the Lov´ asz-Saks construction.) ◮ This talk is not concerned with the Log-Rank conjecture. (It’s trivially true for random matrices).

Lov´ asz-Saks construction Properties of the Lov´ asz-Saks construction: Bipartite graph H Matrix M Rectangle graph G ⊠ edges 1-entries vertices biclique covering n ◦ rectangle covering n ◦ chromatic n ◦ rc χ incl-wise max incl-wise max incl-wise max biclique 1-rectangle indep set cross-free matching “fooling set” (CC) clique