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Generalized Sparsest Cut and Embeddings of Negative-Type Metrics. Shuchi Chawla, Anupam Gupta, Harald R ä cke Carnegie Mellon University 1/25/05. capacity of cut links demand across cut. Sparsity of a cut =. Finding Bottlenecks.

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generalized sparsest cut and embeddings of negative type metrics

Generalized Sparsest Cut and Embeddings of Negative-Type Metrics

Shuchi Chawla, Anupam Gupta, Harald Räcke

Carnegie Mellon University

1/25/05

finding bottlenecks
capacity of cut links

demand across cut

Sparsity of a cut =

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

10

0.1

Sparsest Cut and Embeddings of Negative-type Metrics

the generalized sparsest cut problem
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

Sparsest Cut and Embeddings of Negative-type Metrics

what s known
What’s known
  • Uniform-demands – a special case

D(x,y) = 1 for all x  y

    • O(log n)-approx [Leighton Rao’88]

based on LP-rounding

    • Cannot do better than O(log n) using the LP
    • O(log n)-approx [Arora Rao Vazirani’04]

based on an SDP relaxation

  • General case
    • O(log n)-approx [Linial London Rabinovich’95 Aumann Rabani’98]

based on LP-rounding and low-distortion embeddings

    • Our result: O(log¾n)-approx

Extends [ARV04] using the same SDP

Sparsest Cut and Embeddings of Negative-type Metrics

a metrics perspective
(d) = ec(e) d(e)

x,y D(x,y) d(x,y)

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) = ec(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

Sparsest Cut and Embeddings of Negative-type Metrics

a metrics perspective1
(d) = ec(e) d(e)

x,y D(x,y) d(x,y)

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]

Sparsest Cut and Embeddings of Negative-type Metrics

a metrics perspective2
(d) = ec(e) d(e)

x,y D(x,y) d(x,y)

Squared-Euclidean, or ℓ2-metrics

2

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

ℓ2

ℓ1

2

Sparsest Cut and Embeddings of Negative-type Metrics

arora et al s o log n approx
ℓ2

2

ℓ2

2

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) > const.  diameter

Implies an avg. distortion of O(log n)

Sparsest Cut and Embeddings of Negative-type Metrics

1 ensuring well spreading
1. Ensuring well-spreading
  • Divide pairs into groups based on distances

Di = { (x,y) : 2i d(x,y)  2i+1 }

  • 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

Sparsest Cut and Embeddings of Negative-type Metrics

gluing the groups
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)

Sparsest Cut and Embeddings of Negative-type Metrics

2 average to worst case distortion
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:
    • Don’t ignore low-distortion pairs completely – keep them around and reduce their importance

Sparsest Cut and Embeddings of Negative-type Metrics

weighting and watching
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 < 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

Sparsest Cut and Embeddings of Negative-type Metrics

summarizing
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

ℓ2

ℓ1

Sparsest Cut and Embeddings of Negative-type Metrics

recent developements
ℓ2

2

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

ℓ1

ℓ2

ℓ1

Sparsest Cut and Embeddings of Negative-type Metrics

open problems
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 don’t seem to work

Sparsest Cut and Embeddings of Negative-type Metrics

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