[PPT] - Rainbow matchings Existence and counting Guillem Perarnau PowerPoint Presentation

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Rainbow matchings

Existence and counting Guillem Perarnau

Universitat Polit` ecnica de Catalunya Departament de Matem` atica Aplicada IV

2nd September 2011 Budapest

joint work with Oriol Serra

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Outline

1

The problem

2

Counting with the Local Lemma

3

Our Approach

4

Random Models

Guillem Perarnau MA4-UPC Rainbow matchings: Existence and counting 2 / 19

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Outline

1

The problem

2

Counting with the Local Lemma

3

Our Approach

4

Random Models

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Rainbow matchings and Latin transversals

Edge coloring. C : E(Kn,n) − → N

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Rainbow matchings and Latin transversals

Edge coloring. C : E(Kn,n) − → N Perfect matching: M = {ei indep}

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Rainbow matchings and Latin transversals

Edge coloring. C : E(Kn,n) − → N Perfect matching: M = {ei indep} Rainbow matching: no repeated colors in M.

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Rainbow matchings and Latin transversals

Edge coloring. C : E(Kn,n) − → N Perfect matching: M = {ei indep} Rainbow matching: no repeated colors in M. Integer square matrix A = {aij} B B @ 1 5 4 2 7 2 6 3 5 4 2 1 3 5 3 1 1 C C A

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Rainbow matchings and Latin transversals

Edge coloring. C : E(Kn,n) − → N Perfect matching: M = {ei indep} Rainbow matching: no repeated colors in M. Integer square matrix A = {aij} Transversal Tσ = {aiσ(i)} B B @ 1 5 4 2 7 2 6 3 5 4 2 1 3 5 3 1 1 C C A

Guillem Perarnau MA4-UPC Rainbow matchings: Existence and counting 4 / 19

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Rainbow matchings and Latin transversals

Edge coloring. C : E(Kn,n) − → N Perfect matching: M = {ei indep} Rainbow matching: no repeated colors in M. Integer square matrix A = {aij} Transversal Tσ = {aiσ(i)} Latin Transversal: no repeated entries in Tσ. B B @ 1 5 4 2 7 2 6 3 5 4 2 1 3 5 3 1 1 C C A

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Open problems on Latin squares - Existence

Every latin square of odd order admits a latin transversal. Conjecture (Ryser, 1967)

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Open problems on Latin squares - Existence

Every latin square of odd order admits a latin transversal. Conjecture (Ryser, 1967) Every latin square admits a partial latin transversal of size n − 1. Conjecture (Brualdi, 1975)

Guillem Perarnau MA4-UPC Rainbow matchings: Existence and counting 5 / 19

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Open problems on Latin squares - Existence

Every latin square of odd order admits a latin transversal. Conjecture (Ryser, 1967) Every latin square admits a partial latin transversal of size n − 1. Conjecture (Brualdi, 1975) Every latin square admits a partial latin transversal of size n − O(log2 n). Theorem (Hatami and Shor, 2008)

Guillem Perarnau MA4-UPC Rainbow matchings: Existence and counting 5 / 19

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Open problems on Latin squares - Existence

Every latin square of odd order admits a latin transversal. Conjecture (Ryser, 1967) Every latin square admits a partial latin transversal of size n − 1. Conjecture (Brualdi, 1975) Every latin square admits a partial latin transversal of size n − O(log2 n). Theorem (Hatami and Shor, 2008) For every integer matrix, if no entry appears more than

n 4e times, then it

has a latin transversal. Proposition (Erd˝

s and Spencer, 1991)

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Open problems on Latin squares - Counting

Let zn be the number of latin transversals of the cyclic group of order n. Then there exists two constants 0 < c1 < c2 < 1 such that cn

1n! < zn < cn 2n!

Conjecture (Vardi, 1991)

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Open problems on Latin squares - Counting

Let zn be the number of latin transversals of the cyclic group of order n. Then there exists two constants 0 < c1 < c2 < 1 such that cn

1n! < zn < cn 2n!

Conjecture (Vardi, 1991) Let zn be the number of latin transversals of the cyclic group of order n. Then an < zn < bn√ nn! where a = 3.246 and b = 0.614. Theorem (McKay, McLeod and Wanless, 2006 / Cavenagh and Wan- less, 2010)

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Outline

1

The problem

2

Counting with the Local Lemma

3

Our Approach

4

Random Models

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Poisson Paradigm

A1, . . . , Am bad events Pr(Ai) = pi, Pr m \

i=1

Ai ! ?

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Poisson Paradigm

A1, . . . , Am bad events Pr(Ai) = pi, Pr m \

i=1

Ai ! ?

1

If Ai are mutually independent Pr m \

i=1

Ai ! =

m

Y

i=1

(1 − pi) ∼ e−µ µ =

m

X

i=1

pi

expected number

f bad events

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Poisson Paradigm

A1, . . . , Am bad events Pr(Ai) = pi, Pr m \

i=1

Ai ! ?

1

If Ai are mutually independent Pr m \

i=1

Ai ! =

m

Y

i=1

(1 − pi) ∼ e−µ µ =

m

X

i=1

pi

expected number

f bad events

2

If µ < 1, by the union bound Pr m \

i=1

Ai ! ≥ 1 −

m

X

i=1

Pr(Ai) = 1 − µ > 0

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Poisson Paradigm

A1, . . . , Am bad events Pr(Ai) = pi, Pr m \

i=1

Ai ! ?

1

If Ai are mutually independent Pr m \

i=1

Ai ! =

m

Y

i=1

(1 − pi) ∼ e−µ µ =

m

X

i=1

pi

expected number

f bad events

2

If µ < 1, by the union bound Pr m \

i=1

Ai ! ≥ 1 −

m

X

i=1

Pr(Ai) = 1 − µ > 0 Poisson paradigm: If the dependencies among Ai are rare. Pr m \

i=1

Ai ! = (1 + o(1))e−µ

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Lov´ asz Local Lemma

dependency graph H, V(H) = {A1, . . . , Am}

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Lov´ asz Local Lemma

dependency graph H, V(H) = {A1, . . . , Am} E(H) = {dependencies among events}

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Lov´ asz Local Lemma

dependency graph H, V(H) = {A1, . . . , Am} E(H) = {dependencies among events}, Pr(Ai | T

j∈S Aj) = Pr(Ai)

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Lov´ asz Local Lemma

dependency graph H, V(H) = {A1, . . . , Am} E(H) = {dependencies among events}, Pr(Ai | T

j∈S Aj) = Pr(Ai)

∃x1, . . . , xm ∈ (0, 1) such that Pr(Ai) ≤ xi Y

Aj ∈N(Ai )

(1 − xj)

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Lov´ asz Local Lemma

dependency graph H, V(H) = {A1, . . . , Am} E(H) = {dependencies among events}, Pr(Ai | T

j∈S Aj) = Pr(Ai)

∃x1, . . . , xm ∈ (0, 1) such that Pr(Ai) ≤ xi Y

Aj ∈N(Ai )

(1 − xj) Then, Pr m \

i=1

Ai ! > 0 EXISTENCE

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Lov´ asz Local Lemma

dependency graph H, V(H) = {A1, . . . , Am} E(H) = {dependencies among events}, Pr(Ai | T

j∈S Aj) = Pr(Ai)

∃x1, . . . , xm ∈ (0, 1) such that Pr(Ai) ≤ xi Y

Aj ∈N(Ai )

(1 − xj) Then, Pr m \

i=1

Ai ! >

m

Y

i=1

(1 − xi) COUNTING (lower bound)

Guillem Perarnau MA4-UPC Rainbow matchings: Existence and counting 9 / 19

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Lov´ asz Local Lemma

dependency graph H, V(H) = {A1, . . . , Am} E(H) = {dependencies among events}, Pr(Ai | T

j∈S Aj) = Pr(Ai)

∃x1, . . . , xm ∈ (0, 1) such that Pr(Ai) ≤ xi Y

Aj ∈N(Ai )

(1 − xj) Then, Pr m \

i=1

Ai ! >

m

Y

i=1

(1 − xi) COUNTING (lower bound) Lopsided version (Erd˝

s and Spencer, 1991)

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Upper bound using local Lemma (Lu and Szekely, 2009)

ε-near dependency graph H, V(H) = {A1, . . . , Am}

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Upper bound using local Lemma (Lu and Szekely, 2009)

ε-near dependency graph H, V(H) = {A1, . . . , Am} Pr(Ai ∩ Aj) = 0 if (i, j) ∈ E(H)

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Upper bound using local Lemma (Lu and Szekely, 2009)

ε-near dependency graph H, V(H) = {A1, . . . , Am} Pr(Ai ∩ Aj) = 0 if (i, j) ∈ E(H) for any S ⊆ [m] \ N(Ai) Pr(Ai | \

j∈S

Aj) ≥ (1 − ε) Pr(Ai)

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Upper bound using local Lemma (Lu and Szekely, 2009)

ε-near dependency graph H, V(H) = {A1, . . . , Am} Pr(Ai ∩ Aj) = 0 if (i, j) ∈ E(H) for any S ⊆ [m] \ N(Ai) Pr(Ai | \

j∈S

Aj) ≥ (1 − ε) Pr(Ai) Then, Pr m \

i=1

Ai ! ≤ Y

i

(1 − (1 − ε) Pr(Ai)) COUNTING (upper bound)

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Matchings of Kn,n

Let M be a collection of matchings of Kn,n (or Kn). Choose a perfect matching F u.a.r.. For any M ∈ M define, AM = {M ⊆ F}

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Matchings of Kn,n

Let M be a collection of matchings of Kn,n (or Kn). Choose a perfect matching F u.a.r.. For any M ∈ M define, AM = {M ⊆ F}

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Matchings of Kn,n

Let M be a collection of matchings of Kn,n (or Kn). Choose a perfect matching F u.a.r.. For any M ∈ M define, AM = {M ⊆ F} V(H) = {AM}M∈M E(H) between AM and AN if M and N are in conflict (M ∪ N is not a matching)

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Matchings of Kn,n

Let M be a collection of matchings of Kn,n (or Kn). Choose a perfect matching F u.a.r.. For any M ∈ M define, AM = {M ⊆ F} V(H) = {AM}M∈M E(H) between AM and AN if M and N are in conflict (M ∪ N is not a matching) Let M be a sparse set of matchings of Kn,n (or Kn). Then H is both a negative dependency graph and an ε-near dependency graph. Theorem (Lu and Szekely, 2009)

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Matchings of Kn,n

Let M be a collection of matchings of Kn,n (or Kn). Choose a perfect matching F u.a.r.. For any M ∈ M define, AM = {M ⊆ F} V(H) = {AM}M∈M E(H) between AM and AN if M and N are in conflict (M ∪ N is not a matching) Let M be a sparse set of matchings of Kn,n (or Kn). Then H is both a negative dependency graph and an ε-near dependency graph. Theorem (Lu and Szekely, 2009)

k-cycle free permutations
Latin rectangles n × r
d-regular graphs (configuration model)

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Outline

1

The problem

2

Counting with the Local Lemma

3

Our Approach

4

Random Models

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Setting the bad events

Set the following bad events Given an edge coloring

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Setting the bad events

Set the following bad events Given an edge coloring Ae,f if e and f independent and same color

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Setting the bad events

Set the following bad events Given an edge coloring Ae,f if e and f independent and same color In this matching A(11),(34) holds

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Setting the bad events

Set the following bad events Given an edge coloring Ae,f if e and f independent and same color In this matching A(11),(34) holds If no event holds, we have a rainbow matching

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Setting the bad events

Set the following bad events Given an edge coloring Ae,f if e and f independent and same color In this matching A(11),(34) holds If no event holds, we have a rainbow matching M = {Ae,f : e, f ∈ E(Kn,n), c(e) = c(f), e and f independent}

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Setting the bad events

Set the following bad events Given an edge coloring Ae,f if e and f independent and same color In this matching A(11),(34) holds If no event holds, we have a rainbow matching M = {Ae,f : e, f ∈ E(Kn,n), c(e) = c(f), e and f independent} but... M is not sparse

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Counting rainbow matchings

Fix an edge–coloring of Kn,n such that no color appears more than n/k

times. Let µ = |M|/n(n − 1).

If k ≥ 12 then there exist two constants γ1(k) < 1 < γ2(k), such that e−γ2(k)µ ≤ Pr(M rainbow) ≤ e−γ1(k)µ for a random matching M. Theorem

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Counting rainbow matchings

Fix an edge–coloring of Kn,n such that no color appears more than n/k

times. Let µ = |M|/n(n − 1).

If k ≥ 12 then there exist two constants γ1(k) < 1 < γ2(k), such that e−γ2(k)µ ≤ Pr(M rainbow) ≤ e−γ1(k)µ for a random matching M. Theorem If zn is the number of latin transversals in a latin square of size n where each entry appear at most n/k times (k ≥ 12), then there exist 0 < c1 < c2 < 1 constants such that cn

1n! ≤ zn ≤ cn 2n!

Corollary

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Outline

1

The problem

2

Counting with the Local Lemma

3

Our Approach

4

Random Models

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