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Generalized Sparsest Cut and Embeddings of Negative-Type 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

capacity of cut links

demand across cut

Sparsity of a cut =

- 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)

xS, yS D(x,y)

- Sparsity of graph G,
(G) = minSV(S)

- Our result: an O(log¾n)-approximation for (G)

V

S\V

Sparsest Cut and Embeddings of Negative-type 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

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

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

Sparsest Cut and Embeddings of Negative-type Metrics

(d) = ec(e) d(e)

x,y D(x,y) d(x,y)

- 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

(d) = ec(e) d(e)

x,y D(x,y) d(x,y)

- 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

(d) = ec(e) d(e)

x,y D(x,y) d(x,y)

Squared-Euclidean, or ℓ2-metrics

2

- 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

ℓ2

2

ℓ2

2

- 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

- 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

- 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(alog 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)

- “measured-descent” by Krauthgamer, Lee, Mendel & Naor (2004)

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

- 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

- 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

ℓ2

2

- 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

- Khot & Vishnoi:W(log log log n) integrality gap for the SDP
- 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

- 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

Questions?