Yet another algorithm for dense max cut - go greedy

Claire Mathieu

Warren Schudy (presenting)

Brown University Computer Science

SODA 2008

Max cut

• Splitting an area code in two…

• …to maximize long distance charges!

• 2-layer circuit board layout

• Research platform – e.g. first use of SDP in approximation algorithms

Standard greedy for Max-cut

1 0

0 1

2 1

• 0.5-approx for general graphs

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Dense graphs• Definition: (n vertices)• Poly-Time Approximation Schemes for dense

graphs by:– Arora, Karger and Karpinski 95– Fernandez de la Vega 96– Goldreich, Goldwasser and Ron 98.– Frieze and Kannan 99

• We prove the same theorem using a simpler algorithm

2# nedges

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• Take a random sample of vertices• For all colorings of sampled vertices

– Add remaining vertices greedily in random order

• Return best overall coloring foundOPT

Seeded greedy algorithm2

0 1 t

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Constructed coloring

1 0

0 1

0 1

2 21 2

02tAnalyze when it

guesses OPT

Our results• Seeded greedy algorithm satisfies

in time .

• The standard seedless greedy, when repeated times with random order, also works.

• Simpler proof than Alon, Fernandez de la Vega, Kannan, and Karpinski (2003) that the sample complexity of MaxCut is

• Results extend to weighted MAX-r-CSP

2nOPTCUT 21 2

2 nO

)(221poly

41~ O

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Talk outline• Introduction (done)

• Analysis of seeded greedy:– Introduction of the smoothed coloring– Using the relation between the smoothed and

constructed colorings to lower-bound the number of cut edges (profit) of the output

• Conclusions

Before choosing a random vertex, determine the

greedy color for each

Are we done updating S? No, because 1/3 of C was greedy, but only 1/7 of S was

greedy!

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Constructed coloring C

1 0

0 1

0 0

1 11 0

The Smoothed ColoringSmoothed coloring S(initialized to OPT)Time: 2 2 ½ 3

G

G

G

G

G

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0 1

0 1

1 21 1

Next vertex…

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• Update:

Sg(v)v

SCg(u)uu

Cg(v)v

to wedgea add , vertex dunprocesse still

and both in with color vertex,dunprocesse random

?in color greedy is what , vertex dunprocesse

Constructed coloring (C)

Smoothed coloring (S)

Time: 3 3 ½ 4G

G

G

G

4

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0 1

1 21 1

Another vertex…

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Constructed coloring (C)

Smoothed coloring (S)

Time: 4 ½ 5

G G

G

• Update:

Sg(v)v

SCg(u)uu

Cg(v)v

to wedgea add , vertex dunprocesse still

and both in with color vertex,dunprocesse random

?in color greedy is what , vertex dunprocesse

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2 21 1

Penultimate

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Constructed coloring (C)

Smoothed coloring (S)

Time: 65

G G

• Update:

Sg(v)v

SCg(u)uu

Cg(v)v

to wedgea add , vertex dunprocesse still

and both in with color vertex,dunprocesse random

?in color greedy is what , vertex dunprocesse

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1 2

Final vertex

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• Smoothed coloring starts at OPT and ends at output

• Therefore it suffices to bound the change in profit of the smoothed coloring at each time step

Constructed coloring (C)

Smoothed coloring (S)

Time: 76

G

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S Changes Slowly

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Smoothed coloring (S)Time: 4 Time: 5

• At most (fractional) vertices change color

• Consider each changing vertex separately (interactions negligible).

nttn 1)(1

This vertex will gain a blue wedge and becomes . Net

change:into

Bounding the lost profit

031121''hereprofit lost rbblueLv

(Blue wins ties)

3

51r

3

51b

3/5'r 3/4'b

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factor scaling

a is t

n

3

5

Constructed coloring (C)

Smoothed coloring (S)

vVertex

.0 so blue, choseGreedy br

Time: 3

Finishing the proof

)(/1

2

lossprofit Overall

/ t at time loss Overall

/'' :argument Martingale

''''

'0')'()()'(''

2

2

2n

/1t2/3

2

2

nOn

t

n

CUTOPT

tnOt

nOLE

tnObbrrE

rrbbbluerbblueL

rrbbrrrbbbrb

vv

v

tnO

bluevertices

/

samples

using quantity

a estimatingError

t

n

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By greedy

Conclusions

• Problem: dense weighted max cut and max-CSP

• Algorithm: seeded greedy

• Analysis:– Smoothed / extrapolated coloring– Martingale

• Bonus: simpler sample complexity proof

Questions?• Acknowledgments:

– Brown theory lunch and Claire Mathieu for comments on preliminary talks.

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