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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

Shuchi Chawla, Anupam Gupta, Harald Räcke

Carnegie Mellon University

1/25/05

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 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 Metrics

• 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

( Metricsd) = 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

( Metricsd) = 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

( Metricsd) = 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

Metrics2

2

ℓ2

2

Arora et al.’s O(log n)-approx

• 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

• 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

• 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 Metrics

• 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

• 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 Metrics

• 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… Metrics

• 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

Metrics2

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 Metrics

• 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