randomized coloring with symmetry breaking › ~sriram › talks › randomcoloring.pdf · each...

IntroductionFrugal ColoringMain Theorem

Showing Concentration

Randomized Coloring with Symmetry Breaking

Sriram V. Pemmaraju1

1Department of Computer ScienceThe University of Iowa

[email protected]/∼sriram/

MPII, Saarbrücken

Sriram V. Pemmaraju Randomized Coloring with Symmetry Breaking

Randomized Coloring Procedure

COLORING STEP. Each vertex v picks a tentative coloruniformly at random from its palette.

REPAIR STEP. The tentative coloring may be improper; itis repaired by uncoloring each vertex v that receives thesame color as a neighbor.

The coloring may be completed deterministically (e.g.,greedily).

Iterative Randomized Coloring Procedure

One application of RCP yields a “good,” partial coloringwith positive (though, usually very small) probability.

A “good” coloring is fixed and then extended by performingthe next iteration of the RCP on the uncolored graph.

This is an example of the semi-random method or Rödl Nibble.

Success Stories

Kim (CPC, 1995)

If girth(G) ≥ 5 then

χ(G) ≤ ∆

ln ∆(1 + o(1)).

Success Stories

Johansson (1996)

There exists a constant ∆0: if girth(G) ≥ 4 and ∆ ≥ ∆0 then

χ(G) ≤ 9∆

ln ∆.

Success Stories

See “Graph Coloring and the Probabilistic Method” by Molloyand Reed for more examples.

Symmetry-Breaking Augmentation

Pick π, a permutation of the vertices.

Depending on the application, π can be chosen arbitrarily,randomly, or depending on the structure of the graph.

In the “repair” step, a vertex v is uncolored only if there is aneighbor of lower rank in π that has received the sametentative color.

How does this help?

Key step in analyzing RCP is showing the concentration ofcertain random variables.

Concentration inequalities such as Azuma’s MartingaleInequalityare often used.

Symmetry-breaking makes it easier to apply Azuma’sInequality.

How does this help?

Azuma’s Martingale Inequality

LemmaLet X be a random variable determined by n trialsT1, T2, . . . , Tn, such that for each i, and any two possiblesequences of outcomes t1, t2, . . . , ti−1, ti and t1, t2, . . . , ti−1, ti ′:∣∣∣E [X |T1 = t1, . . . , Ti = ti ]− E [X |T1 = t1, . . . , Ti = ti ′]

∣∣∣ ≤ ci (1)

thenPr

[|X − E [X ]| > t

]≤ 2e−t2/(2

i ). (2)

Difficulty Applying Azuma’s Inequality

Pr[|X − E [X ]| > t

]≤ 2e−t2/(2

Showing ci = O(1) only leads to the bound e−ct2/n forsome positive constant c.

What is desired is eΩ(t2/E [X ]).

Then setting t =√

E [X ] · log E [X ] yields a sharpconcentration.

Pr[|X − E [X ]| > t

]≤ 2e−t2/(2

Then setting t =√

Pr[|X − E [X ]| > t

]≤ 2e−t2/(2

Then setting t =√

Pr[|X − E [X ]| > t

]≤ 2e−t2/(2

Then setting t =√

Applications of Symmetry-Breaking

In proving existence of and deriving algorithms for frugalcolorings.

In proving existence of and deriving algorithms forweighted equitable colorings.

Applications of Symmetry-Breaking

In proving existence of and deriving algorithms for frugalcolorings.

In proving existence of and deriving algorithms forweighted equitable colorings.

Frugal Coloring

DefinitionA β-frugal coloring of a graph G is a proper vertex coloring inwhich no color appears more than β times in a neighborhood.

Lower Bound (Hind, Molloy and Reed, Combinatorica 1997):There are graphs for which every (∆ + 1)-coloring has frugality

(log ∆

log log ∆

Frugal Coloring

DefinitionA β-frugal coloring of a graph G is a proper vertex coloring inwhich no color appears more than β times in a neighborhood.

Lower Bound (Hind, Molloy and Reed, Combinatorica 1997):There are graphs for which every (∆ + 1)-coloring has frugality

(log ∆

log log ∆

Upper Bounds

Hind, Molloy, and Reed (1997): Every graph has a(∆ + 1)-coloring that is O(log5 ∆)-frugal.

This proof can be tightened to yield a (∆ + 1)-coloring thatis O(log3 ∆)-frugal.

This talk

Every graph has a (∆ + 1)-coloring that is O(

log2 ∆log log ∆

)-frugal.

Upper Bounds

This talk

log2 ∆log log ∆

)-frugal.

Upper Bounds

This talk

log2 ∆log log ∆

)-frugal.

Key Elements

Iterative RCP is used, with O(log ∆) iterations.

The symmetry-breaker π is chosen arbitrarily.

Vertices have associated activation probabilities: a vertexv has activation probability av ; it goes to sleep in aniteration with probability 1 − av .

Key Elements

More Details: An Iteration

Initially, each vertex v has palette P(v) = 1, 2, . . . ,∆ + 1.1 Vertex v picks a tentative color uniformly at random from

P(v).2 Vertex v computes its activation probability av and dozes

off with probability 1 − av (independently).3 A vertex that is awake examines the tentative colors picked

by lower ranked neighbors.4 If there is a “color conflict,” the vertex uncolors itself;

otherwise the vertex makes it color permanent.5 Vertex v deletes from P(v) all colors that became

permanent at neighbors during this iteration

Overview of the Proof

Things to prove for each iteration:The iteration is frugal , i.e., each color is used at mostO( log ∆

log log ∆) times.

Sufficient progress is made in the iteration, i.e., maximumdegree shrinks by a constant factor.

Note that |P(v)| ≥ degree(v) + 1.

log log ∆) times.

Palette Size Needs Controlling

With high probability, abouthalf of v ’s neighbors will becolored 1.

log log ∆) times.

Palette size does not shrink too much relative to ∆.

log log ∆) times.

Main Theorem

TheoremSuppose at the beginning of an iteration, we have for all v

|P(v)| ≥ degree(v) + 1 |P(v)| ≥ ∆/4.

Then, for some C ⊆ V, there is a proper list coloring of G[C]such that

1 At most 9 log ∆log log ∆ vertices share a color in any neighborhood.

2 degree C(v) ≤ ∆ ·(1 − 1

)+ 27 ·

√∆ log ∆ for all v

3 degreeC(v) ≤ ∆ · 1e5 + 27 ·

√∆ log ∆ for all v .

Proof of Frugality

Pr[u is tentatively colored x ] =1

|P(u)|.

E[|neighbors of v tentatively colored x |

∑u∈N(v)

1|P(u)|

≤ ∆

∆/4≤ 4.

Proof of Frugality

Since choice of tentative colors is independent, via ChernoffBounds we get

Pr[|neighbors of v tentatively colored x | > 9 log ∆

log log ∆

1∆6 .

Proof of Progress

L(v , x) = set of lower ranked neighbors of v containing x intheir palette. Pr[v is permanently colored] is

av ·∑

x∈P(v)

1|P(v)|

∏u∈L(v ,x)

(1 − 1

|P(u)|

)︸︷︷︸

Understanding qv

qv =∑

x∈P(v)

1|P(v)|

∏u∈L(v ,x)

(1 − 1

|P(u)|

This achieves minimum at

∑x∈P(v)

1|P(v)|

1 − 1∆/4

x∈P(v)

1|P(v)|

· e−5 =1e5 .

Understanding qv

qv =∑

x∈P(v)

1|P(v)|

∏u∈L(v ,x)

(1 − 1

|P(u)|

This can be as high as

∑x∈P(v)

1|P(v)|

· 1 = 1.

For example when L(v , x) = ∅.

The Choice of av

RecallPr[v is permanently colored] = av · qv .

Now set av = 1qv ·e5 . Then,

av ≤ 1 (since qv ≥ 1e5 ) and

Pr[v is permanently colored] = 1e5 .

The Choice of av

Results in Expectation

E [ |neighbors of v permanently colored| ] =degree(v)

We expect v to have

degree(v)

e5 ≤ ∆

colored neighbors and(1 − 1

)· degree(v) ≤

(1 − 1

)·∆

uncolored neighbors.

Azuma’s Martingale Inequality

LemmaLet X be a random variable determined by n trialsT1, T2, . . . , Tn, such that for each i, and any two possiblesequences of outcomes t1, t2, . . . , ti−1, ti and t1, t2, . . . , ti−1, ti ′:∣∣∣E [X |T1 = t1, . . . , Ti = ti ]− E [X |T1 = t1, . . . , Ti = ti ′]

∣∣∣ ≤ ci (3)

thenPr

[|X − E [X ]| > t

]≤ 2e−t2/(2

i ). (4)

Applying Azuma’s Inequality

X = number of neighbors of v that are permanently colored.Each trial Ti is the choice of

tentative color andwhether to stay awake

at a vertex u in the 2-neighborhood of v .

Number of trials can be as large as ∆2.

Temporarily ignore symmetry-breaking.Arbitrarily, order and label vertices in the 2-neighborhoodof v : u1, u2, . . . , uk .Ti is the choices made at ui .Target for t ∼ c ·

√∆ log ∆. So

e−t2/(2P

c2i ) = e−c2∆ log ∆/(2

i ) = ∆−c2∆/(2P

c2i ).

If we can show that∑

c2i = O(∆), we would get a 1

upper bound, by choosing constant c large enough.

√∆ log ∆. So

e−t2/(2P

c2i ) = e−c2∆ log ∆/(2

i ) = ∆−c2∆/(2P

c2i ).

√∆ log ∆. So

e−t2/(2P

c2i ) = e−c2∆ log ∆/(2

i ) = ∆−c2∆/(2P

c2i ).

√∆ log ∆. So

e−t2/(2P

c2i ) = e−c2∆ log ∆/(2

i ) = ∆−c2∆/(2P

c2i ).

Recall,∣∣∣E [X |T1 = t1, . . . , Ti = ti ]− E [X |T1 = t1, . . . , Ti = ti ′]∣∣∣ ≤ ci

Note that∑

c2i may have ∆2 terms. To show

i = O(∆), it is

not enough to show to show ci = O(1).

Upper Bound on Expected Difference

We want to upper bound:∣∣∣E [X |T1 = t1, . . . , Ti = ti ]− E [X |T1 = t1, . . . , Ti = ti ′]∣∣∣

Bad example: expecteddifference can be as much asdegree(v).

Symmetry-Breaking in Action

Order 2-neighborhood vertices by increasing rank in π.Reconsider∣∣∣E [X |T1 = t1, . . . , Ti = ti ]− E [X |T1 = t1, . . . , Ti = ti ′]

∣∣∣Changing T = ti to Ti = t ′i cannot affect verticesu1, u2, . . . , ui−1.It can only affect vertices ui+1, ui+2, . . . , un, but thesevertices affect X only in expectation.

Concentration Result

Using these observations we show that∑

c2i ≤ 81 ·∆.

LemmaLet Pv denote the number of neighbors of v that arepermanently colored in this iteration.

Pr[∣∣∣Pv −

degree(v)

∣∣∣ > 27 ·√

∆ log ∆

2∆4.5 .

Proof Sketch

Si = uj | j > i and uj is adjacent to both v and ui.

These are the only vertices that are affected by the changein the tentative color at ui from ti to t ′i and can in turn affectX .

But only for those uj ∈ Si that tentatively choose color ti ort ′i .

Proof Sketch

Pr[uj chooses tj or t ′j ] =2

|P(uj)|≤ 8

Therefore, by linearity of expectation, the expected numberof vertices in Si that choose ti or t ′i is at most |Si |/8∆.

Expected difference is bounded above by ci = |Si |/8∆.

Proof Sketch

|P(uj)|≤ 8

Proof Sketch

|P(uj)|≤ 8

Proof Sketch

We use inequalities:1 |Si | ≤ ∆

i |Si | ≤ ∆2

to get ∑i

|Si |2 ≤ ∆ · (∑

|Si |) ≤ ∆3.

Hence, ∑i

c2i ≤

(8|Si |∆

≤ 64∆.

Final Step: Apply LLL

Let Bv denote the “bad event” thatfor some color x , vertex v has more than 9 log ∆

log log ∆ neighborscolored x or|Pv − degree(v)

e4 | > 27 ·√

∆ log ∆.

From the concentration results we get

Pr[Bv ] <(∆ + 1)

∆6 +2

∆4.5 ≤ 4∆4.5 .

Each Bv is mutually independent of all except ∆4 other bad

events.

e4 | > 27 ·√

∆ log ∆.

Pr[Bv ] <(∆ + 1)

∆6 +2

∆4.5 ≤ 4∆4.5 .

events.

e4 | > 27 ·√

∆ log ∆.

Pr[Bv ] <(∆ + 1)

∆6 +2

∆4.5 ≤ 4∆4.5 .

events.

e4 | > 27 ·√

∆ log ∆.

Pr[Bv ] <(∆ + 1)

∆6 +2

∆4.5 ≤ 4∆4.5 .

events.

Other Applications of RCP with Symmetry-Breaking

Frugal colorings of special classes of graphs: d-inductivegraphs, sparse graphs.

Weighted equitable colorings.

Last Slide

This is joint work with Aravind Srinivasan at University ofMaryland.

Thanks to Rajiv Raman for arranging my visit and talk.

Any questions?

Last Slide

Any questions?

Last Slide

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