yet another algorithm for dense max cut - go greedy claire mathieu warren schudy (presenting) brown...
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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
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
(Animation done)
• 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
(Animation done)
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
(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
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!
(Animation done)
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
(Animation done)
0 1
0 1
1 21 1
Next vertex…
(Animation done)
• 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
(Animation done)
0 1
1 21 1
Another vertex…
(Animation done)
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
(Animation done)
2 21 1
Penultimate
(Animation done)
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
(Animation done)
1 2
Final vertex
(Animation done)
• 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
(Animation done)
S Changes Slowly
(Animation done)
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
(Animation done)
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
(Animation done)
By greedy
Conclusions
• Problem: dense weighted max cut and max-CSP
• Algorithm: seeded greedy
• Analysis:– Smoothed / extrapolated coloring– Martingale
• Bonus: simpler sample complexity proof