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

The Rectangle Covering number of Random Boolean Matrices

Dirk Oliver Theis

University of Tartu Estonia

Mozhgan Pourmoradnasseri

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

• f 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

• f blicliques

needed needed to cover all to cover all 1-entries edges in M

• f H

Def.: 1-Rectangle in M: K × L w/ Mk,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” Gn,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” Gn,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” Gn,p

⊠

G

⊠(M)

This talk:

◮ Bound rc(Mn,p) = χ(Gn,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” Gn,p

⊠

G

⊠(M)

This talk:

◮ Bound rc(Mn,p) = χ(Gn,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 Gn,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 Gn,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

V =

• (k, ℓ) | Mk,ℓ = 1
• 

     ∗ ∗ ∗ ∗ ∗ ∗ 1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 1 ∗ ∗ ∗ ∗ ∗ ∗       adjacent

Lov´ asz-Saks Rectangle Graph G

⊠(M)

01 matrix M − → Graph G

⊠(M)

Vertices of G

⊠(M): 1-entries

V =

• (k, ℓ) | Mk,ℓ = 1
• 

     ∗ ∗ ∗ ∗ ∗ ∗ 1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 1 ∗ ∗ ∗ ∗ ∗ ∗       adjacent

Edges of G

⊠(M):

“Two 1-entries (= vertices) are adjacent, if they span a 2 × 2 rectangle containing a 0.” E =

• {(k, ℓ), (k′, ℓ′)} | Mk,ℓMk′,ℓ′ = 1

& Mk′,ℓMk,ℓ′ = 0

Lov´ asz-Saks Rectangle Graph G

⊠(M)

01 matrix M − → Graph G

⊠(M)

Vertices of G

⊠(M): 1-entries

V =

• (k, ℓ) | Mk,ℓ = 1
• 

     ∗ ∗ ∗ ∗ ∗ ∗ 1 ∗ ?? ∗ ∗ ∗ ∗ ∗ ∗ ∗ ?? ∗ 1 ∗ ∗ ∗ ∗ ∗ ∗       adjacent

Edges of G

⊠(M):

“Two 1-entries (= vertices) are adjacent, if they span a 2 × 2 rectangle containing a 0.” E =

• {(k, ℓ), (k′, ℓ′)} | Mk,ℓMk′,ℓ′ = 1

& Mk′,ℓMk,ℓ′ = 0

Lov´ asz-Saks Rectangle Graph G

⊠(M)

01 matrix M − → Graph G

⊠(M)

Vertices of G

⊠(M): 1-entries

V =

• (k, ℓ) | Mk,ℓ = 1
• 

     ∗ ∗ ∗ ∗ ∗ ∗ 1 ∗ 1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ 1 ∗ 1 ∗ ∗ ∗ ∗ ∗ ∗       not adjacent

Edges of G

⊠(M):

“Two 1-entries (= vertices) are adjacent, if they span a 2 × 2 rectangle containing a 0.” E =

• {(k, ℓ), (k′, ℓ′)} | Mk,ℓMk′,ℓ′ = 1

& Mk′,ℓMk,ℓ′ = 0

Lov´ asz-Saks Rectangle Graph G

⊠(M)

01 matrix M − → Graph G

⊠(M)

Vertices of G

⊠(M): 1-entries

V =

• (k, ℓ) | Mk,ℓ = 1
• 

     ∗ ∗ ∗ ∗ ∗ ∗ 1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 1 ∗ 1 ∗ ∗ ∗ ∗ ∗ ∗       adjacent

Edges of G

⊠(M):

“Two 1-entries (= vertices) are adjacent, if they span a 2 × 2 rectangle containing a 0.” E =

• {(k, ℓ), (k′, ℓ′)} | Mk,ℓMk′,ℓ′ = 1

& Mk′,ℓMk,ℓ′ = 0

Lov´ asz-Saks Rectangle Graph G

⊠(M)

01 matrix M − → Graph G

⊠(M)

Vertices of G

⊠(M): 1-entries

V =

• (k, ℓ) | Mk,ℓ = 1
• 

     ∗ ∗ ∗ ∗ ∗ ∗ 1 ∗ 1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 1 ∗ ∗ ∗ ∗ ∗ ∗       adjacent

Edges of G

⊠(M):

“Two 1-entries (= vertices) are adjacent, if they span a 2 × 2 rectangle containing a 0.” E =

• {(k, ℓ), (k′, ℓ′)} | Mk,ℓMk′,ℓ′ = 1

& Mk′,ℓMk,ℓ′ = 0

Lov´ asz-Saks Rectangle Graph G

⊠(M)

01 matrix M − → Graph G

⊠(M)

Vertices of G

⊠(M): 1-entries

V =

• (k, ℓ) | Mk,ℓ = 1
• 

     ∗ ∗ ∗ ∗ ∗ ∗ 1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 1 ∗ ∗ ∗ ∗ ∗ ∗       adjacent

Edges of G

⊠(M):

“Two 1-entries (= vertices) are adjacent, if they span a 2 × 2 rectangle containing a 0.” E =

• {(k, ℓ), (k′, ℓ′)} | Mk,ℓMk′,ℓ′ = 1

& Mk′,ℓMk,ℓ′ = 0

Rectangle Graph — Example

Definition of rectangle graph G

⊠(M):

V :=

• (k, ℓ)
• Mk,ℓ = 1
• = set of 1-entries of M

E : (k, ℓ) ∼ (k′, ℓ′) iff Mk,ℓ′Mk′,ℓ = 0

Example

          1 1 1 1 1 1 1 1          

Rectangle Graph — Example

Definition of rectangle graph G

⊠(M):

V :=

• (k, ℓ)
• Mk,ℓ = 1
• = set of 1-entries of M

E : (k, ℓ) ∼ (k′, ℓ′) iff Mk,ℓ′Mk′,ℓ = 0

Example

          1 1 1 1 1 1 1 1          

Rectangle Graph — Example

Definition of rectangle graph G

⊠(M):

V :=

• (k, ℓ)
• Mk,ℓ = 1
• = set of 1-entries of M

E : (k, ℓ) ∼ (k′, ℓ′) iff Mk,ℓ′Mk′,ℓ = 0

Example

          1 1 1 1 1 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 . . . ... . . . 1 . . . 1     all-1 matrix of appropriate dimensions.

The Log-Rank conjecture

Log-Rank Conjecture, Communication Complexity version

Deterministic CC(M) ≤ polylog rk(M)

Log-Rank Conjecture, Graph Theory version

χ(G) ≤ 2polylog 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

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

Bipartite graph H Matrix M= Adj(H) Rectangle graph G

⊠(M)

Clique U ⊆ V (G

⊠(M))

|U| = r

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

Bipartite graph H Matrix M= Adj(H)

• r 1-entries

Rectangle graph G

⊠(M)

Clique U ⊆ V (G

⊠(M))

|U| = r

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

Bipartite graph H Matrix M= Adj(H)

• r 1-entries
• any 2 of which

span a 2 × 2 rectangle which contains a 0 Rectangle graph G

⊠(M)

Clique U ⊆ V (G

⊠(M))

|U| = r

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

Bipartite graph

H

Matrix

M= Adj(H)

• r 1-entries
• any 2 of which span a 2 × 2

rectangle which contains a 0     1 A B D A 1 C E B C 1 F D E F 1     At least one of the As has to be 0. Same for B,C,D,E,F,. . .

Rectangle graph

G

⊠(M)

Clique U ⊆ V (G

⊠(M))

|U| = r

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

Bipartite graph

H

• “Cross-free

matching”: r independent edges no two of which induce a K2,2.

Matrix

M= Adj(H)

• r 1-entries
• any 2 of which span a 2 × 2

rectangle which contains a 0     1 A B D A 1 C E B C 1 F D E F 1     At least one of the As has to be 0. Same for B,C,D,E,F,. . .

Rectangle graph

G

⊠(M)

Clique U ⊆ V (G

⊠(M))

|U| = r

Lov´ asz-Saks Rectangle Graph G

⊠(M)

Basics on the random graphs Gn,p

⊠

The clique number (= fooling set size) The independence number (= largest 1-rectangle) The chromatic number (= rc, ˆ = NdCC)

The random matrices/graphs

Random matrix:

Mn,p: n × n matrix, entries independent, Mn,p

k,ℓ =

• 1

w/ probability p, w/ probability 1 − p. Random graph Gn,p

⊠

:= G

⊠

• Mn,p

rectangle graph of the random matrix

The random matrices/graphs

Random matrix:

Mn,p: n × n matrix, entries independent, Mn,p

k,ℓ =

• 1

w/ probability p, w/ probability 1 − p. Random graph Gn,p

⊠

:= G

⊠

• Mn,p

rectangle graph of the random matrix As usual: n → ∞, p = p(n)

◮ ω, α, χ known when p = Θ(1) (or p = 1/2) ◮ Not when p = o(1) or p = 1 − o(1).

Number of vertices of Gn,p

⊠

Number N of vertices of Gn,p

⊠

(= number of 1s in M n,p)

◮ Binomial random variable Bin(n2, p) ◮ nice concentration near pn2

BTW: Ω(n−3/2) = p = 1 − Ω(n−3/2)

(We couldn’t see anything interesting happening outside of that range.)

Number of edges of Gn,p

⊠

Number M of edges of Gn,p

⊠

◮ Mean: E M = (1 + o(1))(1 − p2)

pn2

2

• ◮ Density δ := 1 − p2

◮ No very good concentration. . . ◮ . . . particularly when p = 1 − o(1)

Number of edges of Gn,p

⊠

Number M of edges of Gn,p

⊠

◮ Mean: E M = (1 + o(1))(1 − p2)

pn2

2

• ◮ Density δ := 1 − p2

◮ No very good concentration. . . ◮ . . . particularly when p = 1 − o(1)

What’s the problem:

               ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗               

Changing the blue 1 to a 0 may introduce up to (n − 1)2 new edges.

Concentration of clique size and chromatic number

Concentration of ω(Gn,p

⊠ )

ω = E ω + O(√n)

McDiarmid’s ieq no-brainer: Changing a row can change ω by at most 1.

Concentration of clique size and chromatic number

Concentration of ω(Gn,p

⊠ )

ω = E ω + O(√n)

McDiarmid’s ieq no-brainer: Changing a row can change ω by at most 1.

Concentration of χ(Gn,p

⊠ )

χ = E χ + O(√n)

(Same reason.)

Concentration of clique size and chromatic number

Concentration of ω(Gn,p

⊠ )

ω = E ω + O(√n)

McDiarmid’s ieq no-brainer: Changing a row can change ω by at most 1.

Concentration of χ(Gn,p

⊠ )

χ = E χ + O(√n)

(Same reason.)

Bad news:

◮ For p = Ω(1), √n is large compared to E ω = O(ln n) ◮ For p = 1 − o(1), √n will be large compared to

E χ = O((1 − p)n ln n)

Lov´ asz-Saks Rectangle Graph G

⊠(M)

Basics on the random graphs Gn,p

⊠

The clique number (= fooling set size) The independence number (= largest 1-rectangle) The chromatic number (= rc, ˆ = NdCC)

Clique number

Theorem [PT].

Clique number ω(Gn,p

⊠ ):

(a) p = o(1/√n): ω = (1−o(1)) ν(Mn,p)

(with ν := size of largest matching in the bipartite graph with adj. mtx. M)

Clique number

Theorem [PT].

Clique number ω(Gn,p

⊠ ):

(a) p = o(1/√n): ω = (1−o(1)) ν(Mn,p)

(with ν := size of largest matching in the bipartite graph with adj. mtx. M)

(b) p = c/√n: ω ≥ 1 − e−c c · n.

Clique number

Theorem [PT].

Clique number ω(Gn,p

⊠ ):

(a) p = o(1/√n): ω = (1−o(1)) ν(Mn,p)

(with ν := size of largest matching in the bipartite graph with adj. mtx. M)

(b) p = c/√n: ω ≥ 1 − e−c c · n. (c) 1/√n ≪ p Θ(1): ω drops from Θ(n) to Θ(ln n).

Clique number

Theorem [PT].

Clique number ω(Gn,p

⊠ ):

(a) p = o(1/√n): ω = (1−o(1)) ν(Mn,p)

(with ν := size of largest matching in the bipartite graph with adj. mtx. M)

(b) p = c/√n: ω ≥ 1 − e−c c · n. (c) 1/√n ≪ p Θ(1): ω drops from Θ(n) to Θ(ln n). (d) p = 1 − o(1): log1/δ n − 1 ≤ ω ≤ 4 log1/δ n − 1.

(recall δ := (1 − p)2 =density of Gn,p

⊠

)

(a,b,c) Tricks using independence n

• of some associated Erd˝
• s-Renyi graph

(d) UB: Union bound; LB: Greedy algorithm

Clique number

Theorem [PT].

Clique number ω(Gn,p

⊠ ):

(a) p = o(1/√n): ω = (1−o(1)) ν(Mn,p)

(with ν := size of largest matching in the bipartite graph with adj. mtx. M)

(b) p = c/√n: ω ≥ 1 − e−c c · n. (c) 1/√n ≪ p Θ(1): ω drops from Θ(n) to Θ(ln n). (d) p = 1 − o(1): log1/δ n − 1 ≤ ω ≤ 4 log1/δ n − 1.

(recall δ := (1 − p)2 =density of Gn,p

⊠

)

(a,b,c) Tricks using independence n

• of some associated Erd˝
• s-Renyi graph

(d) UB: Union bound; LB: Greedy algorithm

Problems:

◮ The constant in (b) is probably not optimal. ◮ In (c), how does ω decrease?

Lov´ asz-Saks Rectangle Graph G

⊠(M)

Basics on the random graphs Gn,p

⊠

The clique number (= fooling set size) The independence number (= largest 1-rectangle) The chromatic number (= rc, ˆ = NdCC)

The independence number of Gn,p

⊠

Theorem (Park & Szpankowski (2005)).

For p = Θ(1), if e−1/(k+1) < p ≤ e−1/k, Largest 1-rectangle has k rows pkn columns

(or vice versa).

The independence number of Gn,p

⊠

Theorem (Park & Szpankowski (2005)).

For p = Θ(1), if e−1/(k+1) < p ≤ e−1/k, Largest 1-rectangle has k rows pkn columns

(or vice versa).

Theorem [PT].

◮ With p = 1 − λ/n, λ = o(n), λ ≥ C:

α

• Gn,p

⊠

• = (1−o(1)) n

eλ. Proof: Chernoff juggling.

Lov´ asz-Saks Rectangle Graph G

⊠(M)

Basics on the random graphs Gn,p

⊠

The clique number (= fooling set size) The independence number (= largest 1-rectangle) The chromatic number (= rc, ˆ = NdCC)

Chromatic number

Theorem (Sherstov).

For p = 1/2:

◮ n/2 ≤ χ ≤ n. ◮ More careful look: χ = (1−o(1)) n = N

α .

(Recall N := |V (Gn,p

⊠

)| = n

• vertices of rectangle graph)

Chromatic number

Theorem (Sherstov).

For p = 1/2:

◮ n/2 ≤ χ ≤ n. ◮ More careful look: χ = (1−o(1)) n = N

α .

(Recall N := |V (Gn,p

⊠

)| = n

• vertices of rectangle graph)

Similar to, e.g., Erd˝

• s-Renyi random graphs: χ ≤ 2 |V |

α .

Chromatic number

Theorem (Sherstov).

For p = 1/2:

◮ n/2 ≤ χ ≤ n. ◮ More careful look: χ = (1−o(1)) n = N

α .

(Recall N := |V (Gn,p

⊠

)| = n

• vertices of rectangle graph)

Similar to, e.g., Erd˝

• s-Renyi random graphs: χ ≤ 2 |V |

α .

Question:

χ(Gn,p

⊠ ) = O

• N

α(Gn,p

⊠ )

• ?

Fractional chromatic number

N α ≤ χ∗ ≤ χ = O(χ∗ · ln N)

• ↑

independence fractional chromatic

folklore fact

ratio chromatic number number Recall the definition of the fractional chromatic number

◮

Take probability distribution µ on the independent sets

◮

• c(µ) := min

v∈V

P

R∼µ(v ∈ R)

◮

• min

µ

1 c(µ) =: χ∗

Fractional chromatic number

N α ≤ χ∗ ≤ χ = O(χ∗ · ln N)

• ↑

independence fractional chromatic

folklore fact

ratio chromatic number number Recall the definition of the fractional chromatic number

◮

Take probability distribution µ on the independent sets

◮

• c(µ) := min

v∈V

P

R∼µ(v ∈ R)

◮

• min

µ

1 c(µ) =: χ∗

Questions:

χ∗(Gn,p

⊠ ) = O

• N

α(Gn,p

⊠ )

• ?

χ(Gn,p

⊠ ) = O

• χ∗(Gn,p

⊠ )

• ?

For Erd˝

• s-Renyi random graphs:

Yes! Yes!

Random Rectangle Graphs:

Conjecture [PT].

For p = 1 − o(1), χ∗(Gn,p

⊠ ) = o

• χ(Gn,p

⊠ )

• Theorem [PT].

If λ = lno(1) n and p = 1 − λ/n: χ∗(Gn,p

⊠ )= O

ln n ln ln n

• ≪

log2 n − O(1) ≤ χ(Gn,p

⊠ )

Chromatic number

Theorem [PT].

If 4 ln n ≤ λ = o(n) and p = 1 − λ/n: N α(Gn,p

⊠ ) = (1−o(1)) eλ = χ∗(Gn,p ⊠ )

In particular,

(1−o(1)) eλ ≤ χ(Gn,p

⊠ ) = O(ln λ · ln n).

Open problems.

• 1. Conjecture. For p = 1 − o(1), χ∗(Gn,p

⊠ ) = o

• χ(Gn,p

⊠ )

• .

Use other LBs on χ, not based on χ∗.

Open problems.

• 1. Conjecture. For p = 1 − o(1), χ∗(Gn,p

⊠ ) = o

• χ(Gn,p

⊠ )

• .

Use other LBs on χ, not based on χ∗.

• 2. Concentration of χ: Is there a function u s.th.

Is χ(Gn,p

⊠ ) = Θ(u(n)), or even = (1−o(1)) u(n)?

Open problems.

• 1. Conjecture. For p = 1 − o(1), χ∗(Gn,p

⊠ ) = o

• χ(Gn,p

⊠ )

• .

Use other LBs on χ, not based on χ∗.

• 2. Concentration of χ: Is there a function u s.th.

Is χ(Gn,p

⊠ ) = Θ(u(n)), or even = (1−o(1)) u(n)?

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

• 3. Determine (give better bounds) on ω(Gn,p

⊠ ) for Ω(1/√n) = p = o(1)!

Open problems.

• 1. Conjecture. For p = 1 − o(1), χ∗(Gn,p

⊠ ) = o

• χ(Gn,p

⊠ )

• .

Use other LBs on χ, not based on χ∗.

• 2. Concentration of χ: Is there a function u s.th.

Is χ(Gn,p

⊠ ) = Θ(u(n)), or even = (1−o(1)) u(n)?

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

• 3. Determine (give better bounds) on ω(Gn,p

⊠ ) for Ω(1/√n) = p = o(1)!

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

• 4. [Deterministic:] Characterize LS rectangle graphs!

4.a) Is there a “combinatorial” characterization which makes sense?!?

4.b) Can LS rectangle graphs be recognized in Poly Time?

Open problems.

• 1. Conjecture. For p = 1 − o(1), χ∗(Gn,p

⊠ ) = o

• χ(Gn,p

⊠ )

• .

Use other LBs on χ, not based on χ∗.

• 2. Concentration of χ: Is there a function u s.th.

Is χ(Gn,p

⊠ ) = Θ(u(n)), or even = (1−o(1)) u(n)?

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

• 3. Determine (give better bounds) on ω(Gn,p

⊠ ) for Ω(1/√n) = p = o(1)!

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

• 4. [Deterministic:] Characterize LS rectangle graphs!

4.a) Is there a “combinatorial” characterization which makes sense?!?

4.b) Can LS rectangle graphs be recognized in Poly Time?

T h a n k Y o u !

Open problems.

• 1. Conjecture. For p = 1 − o(1), χ∗(Gn,p

⊠ ) = o

• χ(Gn,p

⊠ )

• .

Use other LBs on χ, not based on χ∗.

• 2. Concentration of χ: Is there a function u s.th.

Is χ(Gn,p

⊠ ) = Θ(u(n)), or even = (1−o(1)) u(n)?

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

• 3. Determine (give better bounds) on ω(Gn,p

⊠ ) for Ω(1/√n) = p = o(1)!

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

• 4. [Deterministic:] Characterize LS rectangle graphs!

4.a) Is there a “combinatorial” characterization which makes sense?!?

4.b) Can LS rectangle graphs be recognized in Poly Time?