Generalized Sparsest Cut and Embeddings of NegativeType Metrics

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Generalized Sparsest Cut and Embeddings of
Negative-Type Metrics

• Shuchi Chawla, Anupam Gupta, Harald Räcke
• Carnegie Mellon University
• 1/25/05

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Finding Bottlenecks

• Find the cut across which demand exceeds capacity
  by the largest factor

Sparsest cut
Capacity 2.1 units
Demand 3 units
1
Sparsity of the cut 0.7
0.1
10
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The Generalized Sparsest Cut Problem

• The givens
• a graph G(V,E)
• capacities on edges c(e)
• demands on pairs of vertices D(x,y)
• Sparsity of a cut S ? V,
• ?(S) ??(S)c(e)
• ?x?S, y?S D(x,y)
• Sparsity of graph G,
• ?(G) minS?V ?(S)
• Our result an O(log¾n)-approximation for ?(G)

V
S\V
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Whats known

• Uniform-demands a special case
• D(x,y) 1 for all x ? y
• O(log n)-approx Leighton Rao88
• based on LP-rounding
• Cannot do better than O(log n) using the LP
• O(?log n)-approx Arora Rao Vazirani04
• based on an SDP relaxation
• General case
• O(log n)-approx Linial London Rabinovich95
  Aumann Rabani98
• based on LP-rounding and low-distortion
  embeddings
• Our result O(log¾n)-approx
• Extends ARV04 using the same SDP

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A metrics perspective

• Given set S, define a cut metric
• ?S(x,y) 1 if x and y on different sides of
  cut (S, V-S)
• 0 otherwise
• ?(S) ?e c(e) ?S(e)
• ?x,y D(x,y) ?S(x,y)
• Finding sparsest cut
• ? minimizing above function over all
  metrics
• Typical technique Minimize over class M of
  metrics, with M ? l1, and embed into l1

NP-hard
l1
cut
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A metrics perspective

• Finding sparsest cut
• ? minimizing a(d) over metrics
• Lemma Minimize over a class M to obtain d
• have ?-distortion embedding from d into
• ? ?-approx for sparsest cut

l1
l1

• When M all metrics, obtain O(log n)
  approximation
• Linial London Rabinovich 95, Aumann Rabani
  98
• Cannot do any better Leighton Rao 88

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A metrics perspective

• Finding sparsest cut
• ? minimizing a(d) over metrics
• Lemma Minimize over a class M to obtain d
• have ?-avg-distortion embedding from d
  into
• ? ?-approx for uniform-demands
  sparsest cut

l1
l1

• M negative-type metrics ? O(?log n)
  approx
• Arora Rao Vazirani 04
• Question Can we obtain O(?log n) for
  generalized
• sparsest cut,
• or an O(?log n) distortion embedding from
  into

l1
l2
2
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Arora et al.s O(?log n)-approx

• Solve an SDP relaxation to get the best
  representation
• Key Theorem
• Let d be a well-spread-out metric.
  Then ? m an embedding from d
  into a line, such that,
• - for all pairs (x,y), m(x,y) ? d(x,y)
• - for a constant fraction of (x,y), m(x,y) ? 1
  /O(?log n) d(x,y)
• The general case issues
• Well-spreading does not hold
• Constant fraction is not enough
• Want low distortion for every demand pair.

For a const. fraction of (x,y), d(x,y) gt const. ?
diameter
Implies an avg. distortion of O(?log n)
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1. Ensuring well-spreading

• Divide pairs into groups based on distances
• Di (x,y) 2i ? d(x,y) ? 2i1
• At most O(log n) groups
• Each group by itself is well-spread, by
  definition
• Embed each group individually
• distortion O(?log n) contracting embedding into a
  line for each (assume for now)
• Glue the embeddings appropriately

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Gluing the groups

• Start with an a O(?log n) embedding for each
  scale
• A naïve gluing
• concatenate all the embeddings and renormalize by
  dividing by O(?log n)
• Distortion O(a?log n) O(log n)
• A better gluing lemma
• measured-descent by Krauthgamer, Lee, Mendel
  Naor (2004)
• (Recall the previous talk by James Lee)
• Gives distortion O(?a log n) ? distortion
  O(log¾n)

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2. Average to worst-case distortion

• Arora et al.s guarantee a constant fraction of
  pairs embed with low distortion
• We want every pair should embed with low
  distortion
• Idea Re-embed pairs that have high distortion
• Problem Increases the number of embeddings,
  implying a larger distortion
• A re-weighting solution
• Dont ignore low-distortion pairs completely
  keep them around and reduce their importance

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Weighting-and-watching

• Initialize weight 1 for each pair
• Apply ARV to weighted instance
• For pairs with low-distortion,
• decrease weights by factor of 2
• For other pairs, do nothing
• Repeat until total weight lt 1/k
• Total weight decreases by constant factor every
  time
• O(log k) iterations
• Each individual weight decreases from 1 to 1/k
• Each pair contributes to W(log k) iterations
• Implies low distortion for every pair

A constant fraction of the weight is embed with
low distortion
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Summarizing

• Start with a solution to the SDP
• For every distance scale
• Use ARV04 to embed points into line
• Use re-weighting to obtain good worst-case
  distortion
• Combine distance scales using measured-descent
• In practice
• Write another SDP to find best embedding into
• Use J-L to embed into and then into a
  cut-metric

l2
l1
l2
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Recent developements

• Arora, Lee Naor obtained an O(?log n log log n)
  approximation for sparsest cut
• The improvement lies in a better concatenation
  technique
• Nearly optimal embedding from into
• Evidence for hardness
• Khot Vishnoi W(log log log n) integrality gap
  for the SDP
• l.b. for embedding into
• Chawla, Krauthgamer, Kumar, Rabani Sivakumar
• W(log log n) hardness based on Unique Games
  Conjecture
• Evidence that constant factor approximation is
  not possible
• Other approximations using similar SDP
  relaxations
• Feige, Hajiaghayi Lee O(?log n) approx for
  min-wt. vertex cuts

l2
l1
l1
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Open Problems

• Beating the ALN05 O(?log n log log n)
  approximation
• Can the SDP give a better bound?
• Exploring flow-based techniques
• Closing the gap between hardness and
  approximation
• Other applications of SDP with triangle
  inequalities
• Other partitioning problems
• Directed versions? SDP/LP dont seem to work

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Questions?
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Gluing the groups a naïve approach
l2
l1

• Basic plan embed into first and then into
• A naïve idea concatenate all the embeddings and
  renormalize by dividing by O(?log n)
• No expansion ?(x,y) ? d(x,y), ? x,y
• Contraction ?(x,y) ? 1 /O(?log n) . d(x,y)
  /O(?log n)
• Distortion at most O(log n)
• No better than before We unnecessarily lose
  another factor of O(?log n)

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Gluing via measured descent

• A technique developed by Krauthgamer, Lee, Mendel
  Naor (2004)
• Basic idea
• Consider an expansion measure V(x,y)
• V(x,y) is large if the number of points around x
  grows fast with distance from x for distance
  scales in O(d(x,y))
• Fakcharoenphol, Rao Talwar give an embedding in
  which low V(x,y) implies low-distortion for the
  pair (x,y)
• KLMN Embed pairs with large V(x,y) better, so
  as to correct the skew of FRT03

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Gluing via measured descent

• A technique developed by Krauthgamer, Lee, Mendel
  Naor (2004)
• Basic idea
• KLMN Embed pairs with large V(x,y) better, so
  as to correct the skew of FRT03
• Naïve approach create an embedding for every
  distance scale and concatenate
• KLMN
• create an embedding for every number t of
  points
• to embed a point x, use the distance scale
  corresponding to ball around x containing t
  points
• Large V(x,y) ? more representation for (x,y)
  one for each distance scale lying in O(d(x,y))

• Gives an embedding with distortion O(?a log n)
• a ?log n ? distortion O(log¾n)

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ARV

• Novel technique of rounding SDP
• Give rounding with low AVG distortion from l22
  into l1
• What does this mean about uniform sparsest cut
• The main result and l22 metric with large avg
  distance can be embed with low avg dist.

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An intuition

• Essentially want to do the same embedding as ARV
• However, different distance scales no longer
  have well-spreading property
• Solution treat each distance scale separately
• At most log n distance scales
• Naïve concatenation gives a total factor of log n
• No benefit
• Instead, use a smart concatenation based on
  KLMN
• Details need to be filled in

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A single distance scale

• What do we need in the end?
• Forming a partition
• Partition to probabilistic embedding
• Handling probabilities
• Handling weights

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Putting them together

• Basic idea of KLMN

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The algorithm

• Best embedding can be found through an SDP
• We embed into l2 which can then be embed into l1
  and then into a cut metric