Title: Generalized Sparsest Cut and Embeddings of NegativeType Metrics
1Generalized Sparsest Cut and Embeddings of
Negative-Type Metrics
- Shuchi Chawla, Anupam Gupta, Harald Räcke
- Carnegie Mellon University
- 1/25/05
2Finding 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
3The 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
4Whats 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
5A 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
6A 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
7A 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
8Arora 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)
91. 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
10Gluing 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)
112. 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
12Weighting-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
13Summarizing
- 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
14Recent 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
15Open 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
16Questions?
17Gluing 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)
18Gluing 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
19Gluing 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)
20ARV
- 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.
21An 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
22A single distance scale
- What do we need in the end?
- Forming a partition
- Partition to probabilistic embedding
- Handling probabilities
- Handling weights
23Putting them together
24The 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