On Partitioning Graphs via Single Commodity Flows

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On Partitioning Graphs via Single Commodity Flows

• Lorenzo Orecchia
• UC Berkeley
• Leonard J. Schulman
• Caltech
• Umesh V. Vazirani
• UC Berkeley
• Nisheeth K. Vishnoi
• IBM Delhi work done while visiting UC Berkeley

STOC 2008 , Victoria
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The SPARSEST CUT problem
Given a graph G(V,E) and partition
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The SPARSEST CUT problem
Given a graph G(V,E) and partition
SPARSEST CUT find with minimum
expansion . Applications
Divide-and-Conquer, Image Segmentation, VLSI
design, Clustering. Theoretical Importance
Metric Embeddings, Spectral Methods.
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The SPARSEST CUT problem
Given a graph G(V,E) and partition
SPARSEST CUT find with minimum
expansion . Applications
Divide-and-Conquer, Image Segmentation, VLSI
design, Clustering. Theoretical Importance
Metric Embeddings, Spectral Methods. The
SPARSEST CUT problem is NP-hard.
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Approximation Algorithms for SPARSEST CUT



For a d-regular graph G.
All graphs have been sparsified to
edges.
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Approximation Algorithms for SPARSEST CUT



IN PRACTICE Too slow for massive data
sets. Spectral and heuristics like METIS used
instead.
For a d-regular graph G.
All graphs have been sparsified to
edges.
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Fast Approximation Algorithms



CUT-MATCHING GAME FRAMEWORK FOR COMPUTING APPROX
USING s-t MAXFLOW COMPUTATIONS
For a d-regular graph G.
All graphs have been sparsified to
edges.
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Fast Approximation Algorithms
For a d-regular graph G.
All graphs have been sparsified to
edges.
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Our Contribution
IN KRV CUT-MATCHING GAME FRAMEWORK
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Our Contribution
IN KRV CUT-MATCHING GAME FRAMEWORK
LOWER BOUND No better approx than ?((log
n)1/2) in KRV framework.
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Our Contribution
LOWER BOUND No better approx than ?((log
n)1/2) in KRV framework. Best integrality gap
for SDP is ?(loglog(n)).
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Our Contribution
LOWER BOUND No better approx than ?((log
n)1/2) in KRV framework. Best integrality gap
for SDP is ?(loglog(n)). CUT-MATCHING RIGHT
ABSTRACTION?
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The KRV Cut-Matching Game
CUT PLAYER
MATCHING PLAYER
H0
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The KRV Cut-Matching Game
CUT PLAYER
MATCHING PLAYER
H0
50-50 Cut
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The KRV Cut-Matching Game
CUT PLAYER
MATCHING PLAYER
H0
50-50 Cut
Perfect Matching
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The KRV Cut-Matching Game
CUT PLAYER
MATCHING PLAYER
H1
50-50 Cut
Perfect Matching
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The KRV Cut-Matching Game
CUT PLAYER
MATCHING PLAYER
H1
50-50 Cut
Perfect Matching
50-50 Cut
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The KRV Cut-Matching Game
CUT PLAYER
MATCHING PLAYER
H1
50-50 Cut
Perfect Matching
50-50 Cut
Perfect Matching
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The KRV Cut-Matching Game
CUT PLAYER
MATCHING PLAYER
H2
50-50 Cut
Perfect Matching
50-50 Cut
Perfect Matching

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The KRV Cut-Matching Game
CUT PLAYER
MATCHING PLAYER
H2
50-50 Cut
Perfect Matching
50-50 Cut
Perfect Matching

Go until time T when
GOAL Minimize T
GOAL Maximize T
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The KRV Cut-Matching Game
CUT PLAYER
MATCHING PLAYER
H2
50-50 Cut
Perfect Matching
50-50 Cut
Perfect Matching

Go until time T when
GOAL Minimize T
GOAL Maximize T
KRV there exists a cut strategy achieving T
O((log n)2).
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The KRV Cut-Matching Game
CUT PLAYER STRATEGY
Runs in time c(n) per iteration
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The KRV Cut-Matching Game
CUT PLAYER STRATEGY
Runs in time c(n) per iteration
APPROXIMATION ALGORITHM
Running time t(n) (Tmaxflow c(n))
Approx Ratio t(n)
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The KRV Cut-Matching Game
CUT PLAYER STRATEGY
Runs in time c(n) per iteration
APPROXIMATION ALGORITHM
Running time t(n) (Tmaxflow c(n))
Approx Ratio t(n)
Time to compute s-t maxflow in G
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The KRV Cut-Matching Game
CUT PLAYER STRATEGY
Runs in time c(n) per iteration
APPROXIMATION ALGORITHM
Running time t(n) (Tmaxflow c(n))
Approx Ratio t(n)
KRV strategy has c(n) and t(n)
O((log n)2). TOTAL RUNNING TIME
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Our Version of the Cut-Matching Game

• No Stopping Condition
• Value of Game is

• MODIFIED GAME

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Our Version of the Cut-Matching Game

• No Stopping Condition
• Value of Game is

• MODIFIED GAME
• STILL YIELDS
• APPROX ALGORITHM

Approximation Ratio
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Our Version of the Cut-Matching Game

• No Stopping Condition
• Value of Game is

• MODIFIED GAME
• STILL YIELDS
• APPROX ALGORITHM
• RESULTS

Approximation Ratio
CUT STRATEGY
MATCHING STRATEGY
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Cut Strategies Finding Cuts Quickly
After t iterations, Ht M1, M2, , Mt .
1 charge
Random assignment of charge
1 charge
Explain how we connect the mixing of the walk
with expansion
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Cut Strategies Finding Cuts Quickly
After t iterations, Ht M1, M2, , Mt .
1 charge
Random assignment of charge
1 charge
(xy)/2
x
Mix the charges along the matchings M1, M2, ,
Mt
y
(xy)/2
Explain how we connect the mixing of the walk
with expansion
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Cut Strategies Finding Cuts Quickly
After t iterations, Ht M1, M2, , Mt .
1 charge
Random assignment of charge
1 charge
Explain how we connect the mixing of the walk
with expansion
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Cut Strategies Finding Cuts Quickly
After t iterations, Ht M1, M2, , Mt .
1 charge
Random assignment of charge
1 charge
Explain how we connect the mixing of the walk
with expansion
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Cut Strategies Finding Cuts Quickly
After t iterations, Ht M1, M2, , Mt .
1 charge
Random assignment of charge
1 charge
Mix the charges along the matchings M1, M2, ,
Mt
Explain how we connect the mixing of the walk
with expansion
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Cut Strategies Finding Cuts Quickly
After t iterations, Ht M1, M2, , Mt .
1 charge
Random assignment of charge
1 charge
If cut is small, unbalance remains.
Mix the charges along the matchings M1, M2, ,
Mt
Explain how we connect the mixing of the walk
with expansion
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Cut Strategies Finding Cuts Quickly
Order the vertices according to the final charge
present and cut in half.
S
S
n/2
n/2
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The KRV mixing walk
KRV-walk At round t Lazy random walk
traversing matchings in order.

• .

0
0
M1
M2
1
0
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The KRV mixing walk
KRV-walk At round t Lazy random walk
traversing matchings in order.

• .

0
0
M1
Averaging along M1
M2
1/2
1/2
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The KRV mixing walk
KRV-walk At round t Lazy random walk
traversing matchings in order.

• .

1/4
1/4
M1
Averaging along M2
M2
1/4
1/4
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Sketch of KRV Analysis

• Mixing of P(t) measured by potential function
• If P(t) mixes well, Ht has good expansion.
• Possible to embed Kn in Ht.
• Potential Reduction at every iteration
• Decomposition possible as KRV walks matchings
  in order.
• 4. Cut-finding procedure reduces potential by a
  fixed factor

Mixing due to matching Mt
Yields expander in O((log n)2) rounds
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Why KRV cannot do better
Recall Approximation is
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Why KRV cannot do better
Recall Approximation is
Can KRV get better than O(1) expansion?
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Why KRV cannot do better
Recall Approximation is
S
V-S
Can KRV get better than O(1) expansion?
SUPPOSE Walk on M1, M2, , Mk mixes perfectly
on S and V-S and no edge cross (S,V-S)
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Why KRV cannot do better
Recall Approximation is
Mk1
Now walk mixes perfectly. Expansion is O(1).
S
V-S
Can KRV get better than O(1) expansion?
SUPPOSE Walk on M1, M2, , Mk mixes perfectly
on S and V-S and no edge cross (S,V-S)
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Our Cut Strategy a Different Walk

• IDEA use lazy natural random walk
• ADVANTAGES
• Eliminates bad case possible to get better
  expansion.
• Better handle on expansion through mixing by
  Cheegers Inequality.
• CHALLENGE
• Impossible to decompose potential as in KRV.
• Additional matching modifies all steps of walk.

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Our Cut Strategy a Different Walk

• IDEA use lazy natural random walk
• ADVANTAGES
• Eliminates bad case possible to get better
  expansion.
• Better handle on expansion through mixing by
  Cheegers Inequality.
• CHALLENGE
• Impossible to decompose potential as in KRV.
• Additional matching modifies all steps of walk.

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Modified Walk and Matrix Inequalities
CHALLENGE Impossible to decompose potential as
in KRV. Additional matching modifies all steps
of walk. SOLUTION Use round-robin walk close
to natural walk Apply matrix
inequality
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Modified Walk and Matrix Inequalities
CHALLENGE Impossible to decompose potential as
in KRV. Additional matching modifies all steps
of walk. SOLUTION Use round-robin walk close
to natural walk Apply matrix
inequality
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Modified Walk and Matrix Inequalities
CHALLENGE Impossible to decompose potential as
in KRV. Additional matching modifies all steps
of walk. SOLUTION Use round-robin walk close
to natural walk Apply matrix
inequality
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Modified Walk and Matrix Inequalities
SOLUTION Use round-robin walk close to
natural. Apply matrix inequality. Yields same
potential reduction as KRV. But our walk is
better related to expansion In O((log n)2)
rounds, conductance (1\log n) by Cheeger.
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Modified Walk and Matrix Inequalities
SOLUTION Use round-robin walk close to
natural. Apply matrix inequality. Yields same
potential reduction as KRV. But our walk is
better related to expansion In O((log n)2)
rounds, conductance (1\log n) by
Cheeger. ?(log n) expansion in O((log n)2)
rounds.
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Modified Walk and Matrix Inequalities
SOLUTION Use round-robin walk close to
natural. Apply matrix inequality. Yields same
potential reduction as KRV. But our walk is
better related to expansion In O((log n)2)
rounds, conductance (1\log n) by
Cheeger. ?(log n) expansion in O((log n)2)
rounds.
TIME only polylog factors worse than KRV
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Lower Bound
Matching player yielding against any
Cut player.
No better approximation than O((log n)1/2) in
KRV Cut-Matching game
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Lower Bound Idea
A NAÏVE MATCHING PLAYER Fix a cut (S,V-S). Keep
it as sparse as possible.
S
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Lower Bound Idea
A NAÏVE MATCHING PLAYER Fix a cut (S,V-S). Keep
it sparse.
Cut player plays
S
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Lower Bound Idea
A NAÏVE MATCHING PLAYER Fix a cut (S,V-S). Keep
it sparse.
Cut player plays
S
GAME OVER
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Lower Bound Idea
A NAÏVE MATCHING PLAYER Fix a cut (S,V-S). Keep
it sparse.
Cut player plays
S
GAME OVER
IDEA hedge over many cuts
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Lower Bound Idea

• THE REAL PLAYER - AT START
• Matching player selects log(n) orthogonal 50-50
  cuts in V.
• THE REAL PLAYER - THROUGHOUT THE GAME
• Matching player adds matchings to minimize
  average expansion.

Orthogonal 50-50 cuts Minimum correlation
Cut player cannot kill many cuts at once
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Main Lemma

8 50-50 cut (S,V-S),
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Main Lemma

8 50-50 cut (S,V-S), 9 a perfect matching M,
s.t.
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Main Lemma

8 50-50 cut (S,V-S), 9 a perfect matching M,
s.t.
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Conclusion and Open Problems

• POWER OF CUT-MATCHING GAME
• Simple yet powerful framework for SPARSEST CUT.
• OPEN QUESTION
• Can we use Cut-Matching to get fast (log n)1/2
  approximation?