# Slide 1

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```					 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
10     21

01
• 0.5-approx for general graphs

(Animation done)
Dense graphs
• Definition:  # edges  n  (n vertices)
2

• 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

(Animation done)
Seeded greedy algorithm
• Take a random sample of t0  1  2 vertices
Analyze
• For all 2t colorings of sampled vertices
0                                            when it
guesses
– Add remaining vertices greedily in random order      OPT

• Return best overall coloring found
OPT
01
10
Constructed
coloring                     22
12

01

(Animation done)
Our results
• Seeded greedy algorithm satisfies
CUT  OPT  n 2 in time 2 O 1   n 2.
2

• The standard seedless greedy, when
poly(  )
repeated 1 22         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 O 1  4 
~

• Results extend to weighted MAX-r-CSP

(Animation done)
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
The Smoothed Coloring
Smoothed coloring S
Constructed
coloring C
Time:   2 2½ 3        (initialized to OPT)
00
G
10
G

G                 G
10                             11
Before choosing a
random vertex,
determine the              G
greedy color for                           Are we done updating S? No, because 1/3
01              of C was greedy, but only 1/7 of S was
each
greedy!

(Animation done)
Next vertex…                          Smoothed
coloring (S)

01
Time:   3 3½ 4
Constructed                  G
coloring (C)

G             G
11                         12

01   G
• Update:
  unprocesse vertex v, what is greedy color g(v)in C?
d
 u  random unprocesse vertex,color u with g(u) in both C and S
d
  still unprocesse vertex v, add a wedge g(v) to S
d

(Animation done)
Another vertex…                         Smoothed
coloring (S)

01   Time:   4 4½ 5
Constructed                G
coloring (C)

G           G
11                       12

• Update:
  unprocesse vertex v, what is greedy color g(v)in C?
d
 u  random unprocesse vertex,color u with g(u) in both C and S
d
  still unprocesse vertex v, add a wedge g(v) to S
d

(Animation done)
Penultimate                      Smoothed
coloring (S)
Time:   5 6
Constructed
coloring (C)

G           G
11                      22

• Update:
  unprocesse vertex v, what is greedy color g(v)in C?
d
 u  random unprocesse vertex,color u with g(u) in both C and S
d
  still unprocesse vertex v, add a wedge g(v) to S
d

(Animation done)
Final vertex         Smoothed
coloring (S)
Time:   6 7
Constructed
coloring (C)

G
12

• 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

(Animation done)
S Changes Slowly
Time: 4 Smoothed coloring (S) Time: 5

• At most 1  (n  t )  1 t   n (fractional) vertices change
color
• Consider each changing vertex separately (interactions
negligible).

(Animation done)
5 n
 is a         Bounding the lost profit
3 t
scaling factor                                      Smoothed
Constructed
coloring (C)
Time:    3       coloring (S)

5
r  1                   r'  5 / 3      
This vertex will gain
3                                                          a blue wedge and
5                 b'  4 / 3                             becomes       . Net
b  1                                                                 change:
3                                                                 into
(Blue wins ties)
Vertex v

Lv  lost profit here  blue  b'r '  1 12  1 3  0

Greedy chose blue, so r  b  0.

(Animation done)
Finishing the proof
b'r '  (b'b)  (b  r )  (r  r ' )   b'b  0  r  r '
By greedy
Lv  blue  b'r '  blue   b'b  r  r ' 
 
Martingale argument : E  r 'r  b'b   O n / t
Error estimating a
quantity  n using

Overall loss at time t  E  L   O   O n / t 
        n                            t samples
v
 v         t        
Overall profit loss  OPT  CUT
n
  3/ 2 
n2 2n 2
 O(n 2 )
 blue
vertices
t 1 /  2 t   1/  2
 On / t 

(Animation done)
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

(Animation done)

```
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