1. Generalized Sparsest Cut and Embeddings of Negative-Type Metrics
Shuchi Chawla, Anupam Gupta, Harald Räcke
Carnegie Mellon University
1/25/05

2. Sparsest Cut and Embeddings of Negative-type Metrics
Finding Bottlenecks
Find the cut across which demand exceeds capacity by the largest factor
capacity of cut links
demand across cut
Sparsity of a cut =
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

3. 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

4. Sparsest Cut and Embeddings of Negative-type Metrics
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

5. Sparsest Cut and Embeddings of Negative-type Metrics
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
(d) = e c(e) d(e)
x,y D(x,y) d(x,y)
NP-hard ℓ1
cut
Sparsest Cut and Embeddings of Negative-type Metrics

6. Sparsest Cut and Embeddings of Negative-type Metrics
A metrics perspective
(d) = e c(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

7. Squared-Euclidean, or ℓ2-metrics
A metrics perspective
(d) = e c(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 -avg-distortion embedding from d into
 -approx for “uniform-demands” sparsest cut ℓ1
Squared-Euclidean, or ℓ2-metrics 2 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 2 ℓ1
Sparsest Cut and Embeddings of Negative-type Metrics

8. Arora et al.’s O(log n)-approxℓ2 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. ℓ2 2
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

9. 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

10. Sparsest Cut and Embeddings of Negative-type Metrics
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

11. 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

12. 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

13. Sparsest Cut and Embeddings of Negative-type Metrics
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

14. Sparsest Cut and Embeddings of Negative-type Metrics
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 ℓ2 2 ℓ1
Sparsest Cut and Embeddings of Negative-type Metrics

15. Sparsest Cut and Embeddings of Negative-type Metrics
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

16. Questions?