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Graph Sparsifiers

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Graph Sparsifiers

Nick Harvey University of British Columbia

Based on joint work with Isaac Fung,and independent work of RameshHariharan & DebmalyaPanigrahi

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- Floor joists

Wood Joists

Engineered Joists

- Bridges

Masonry Arch

Truss Arch

- Bones

Human Femur

Robin Bone

- Graphs

Dense Graph

Sparse Graph

Cut Sparsifiers

(Karger ‘94)

- Weighted subgraphs that approximately preserve graph structure

- Input: Undirected graph G=(V,E), weights u : E !R+
- Output: A subgraphH=(V,F) of G with weightsw : F!R+ such that |F| is small and
u(±G(U)) = (1§²) w(±H(U)) 8UµV

weight of edges between U and V\U in G

weight of edges between U and V\U in H

G

H

(Spielman-Teng ‘04)

- Weighted subgraphs that approximately preserve graph structure

- Input: Undirected graph G=(V,E), weights u : E !R+
- Output: A subgraphH=(V,F) of G with weightsw : F!R+ such that |F| is small and
xTLGx = (1 §²) xTLHx 8x2RV

Laplacian matrix of G

Laplacian matrix of H

G

H

Algorithm A for some problem P

Dense Input graph G

Exact/Approx Output

Min s-t cut, Sparsest cut,Max cut, …

(Fast) Sparsification Algorithm

Algorithm Aruns faster on sparse input

Sparse graph H

Approximatelypreserves solution of P

Approximate Output

n = # vertices

m = # edges

c = large constant

~

*: The best algorithm in our paper is due to Panigrahi.

Eliminate most of these

- Can’t sample edges with same probability!
- Idea: [Benczur-Karger ’96]Sample low-connectivity edges with high probability, and high-connectivity edges with low probability

Keep this

[Benczur-Karger ‘96]

- Input: Graph G=(V,E), weights u : E !R+
- Output: A subgraph H=(V,F) with weights w : F!R+

- Choose ½ (= #sampling iterations)
- Choose probabilities { pe:e2E}
- For i=1 to ½
- For each edge e2E
- With probability peAdd e to F Increase we by ue/(½pe)

How should we choosethese parameters?

- E[|F|]·½¢epe
- E[ we ] = ue8e2E
- ) For every UµV, E[ w(±H(U)) ] = u(±G(U))

Goal 1: E[|F|] = O(n log n / ²2)

Goal 2: w(±H(U)) is highly concentrated

- Input: Graph G=(V,E), weights u : E !R+
- Output: A subgraph H=(V,F) with weights w : F!R+

- Choose ½ = O(log n /²2)
- Let pe = 1/“strength” of edge e
- For i=1 to ½
- For each edge e2E
- With probability peAdd e to F Increase we by ue/(½pe)

“strength” is a slightly unusual quantity, but

Fact 3:Can estimate all edge strengths inO(m log3 n) time

“strength” is a slightly unusual quantity

Question:[BK ‘02]Can we use connectivity instead of strength?

- Fact 1: E[|F|] = O(n log n / ²2)
- Fact 2: w(±H(U)) is very highly concentrated
- ) For every UµV, w(±H(U)) = (1 §²) u(±G(U))

- Input: Graph G=(V,E), weights u : E !R+
- Output: A subgraph H=(V,F) with weights w : F!R+

- Choose ½ = O(log2 n /²2)
- Let pe = 1/“connectivity” of e
- For i=1 to ½
- For each edge e2E
- With probability peAdd e to F Increase we by ue/(½pe)

- Fact 1: E[|F|] = O(n log2 n / ²2)
- Fact 2: w(±H(U)) is very highly concentrated
- ) For every UµV, w(±H(U)) = (1 §²) u(±G(U))

- Extra trick:Can shrink |F| to O(n log n / ²2) by using Benczur-Karger to sparsify our sparsifier!

Connectivities are simpler and more natural

) Faster to compute

Fact:Can estimate all edge connectivitiesin O(m + n log n) time [Ibaraki-Nagamochi ’92]

) Useful in other scenarios

Our sampling method has been used to compute sparsifiers in the streaming model [Ahn-Guha-McGregor ’12]

Most Cuts are Big & Easy!

Most cuts hit a huge number of edges) extremely concentrated

)whp, most cuts are close to their mean

Hits only one red edge) poorly concentrated

Hits many red edges) reasonably concentrated

Low samplingprobability

High connectivity

There are few small cuts [Karger ’94],

so probably all are concentrated.

Key Question:Are there few such cuts?

Key Lemma: Yes!

The same cut also hits many green edges) highly concentrated

This masks the poor concentration above

High samplingprobability

Low connectivity

- Notation: kuv = min size of a cut separating u and v
- Main ideas:
- Partition edges into connectivity classesE = E1[E2[ ... Elog nwhere Ei = { e : 2i-1·ke<2i }
- Prove weight of sampled edges that each cuttakes from each connectivity class is about right
- This yields a sparsifier

U

Prove weight of sampled edges that each cuttakes from each connectivity class is about right

- Notation:
- C = ±(U) is a cut
- Ci:= ±(U) ÅEi is a cut-induced set

- Chernoff bounds can analyze each cut-induced set, but…
- Key Question: Are there few small cut-induced sets?

C2

C3

C1

C4

- Lemma: [Karger ’93]
Let G=(V,E) be a graph.

Let K be the edge-connectivity of G. (i.e., global min cut value)

Then, for every ®¸1,|{ ±(U) : |±(U)|·®K }| < n2®.

- Example: Let G = n-cycle.
Edge connectivity is K=2.

Number of cuts of size c = £( nc ).)|{ ±(U) : |±(U)|·®K }| ·O(n2®).

- Our Lemma:Let G=(V,E) be a graph. Fix any BµE.
Suppose ke¸K for all e in B. (kuv = min size of a cut separating u and v)

Then, for every ®¸1,|{ ±(U) ÅB : |±(U)|·®K }| < n2®.

- Karger’s Lemma:the special case B=E and K=min cut.

- Lemma: [Karger ’93]
Let G=(V,E) be a graph.

Let K be the edge-connectivity of G. (i.e., global min cut value)

Then, for every c¸K,|{ ±(U) : |±(U)|·c }| < n2c/K.

²

- Example: Let G = n-cycle.
- Edge connectivity is K=2
- |{ cuts of size c}| < nc

K = ²

< n2c/²

- Our Lemma:Let G=(V,E) be a graph. Fix any BµE.
Suppose ke¸K for all e in B. (kuv = min size of a cut separating u and v)

Then, for every ®¸1,|{ ±(U) ÅB : |±(U)|·®K }| < n2®.

²

- Example: Let G = n-cycle.
- Let B = cycle edges.
- We can take K=2.
- So |{ ±(U) ÅB : |±(U)|·®K }| < n2®.
- |{cut-induced subsets of B induced by cuts of size · c}|·nc

- Input: A graph
- Output: A minimum cut (maybe)
- While graph has 2 vertices
- Pick an edge at random
- Contract it

- End While
- Output remaining edges

- Claim: For any min cut, this algorithm outputs it with probability ¸ 1/n2.
- Corollary: There are · n2 min cuts.

Splitting Off

Replace edges {u,v} and {u’,v} with {u,u’}while preserving edge-connectivity

between all vertices other than v

v

v

u

u

u’

u’

- Input: A graph G=(V,E), and BµE
- Output: A cut-induced subset of B
- While graph has 2 vertices
- If some vertex v has no incident edges in B
- Split-off all edges at v and delete v

- Pick an edge at random
- Contract it

- If some vertex v has no incident edges in B
- End While
- Output remaining edges in B

Wolfgang Mader

- Claim: For any min cut-induced subset of B, this algorithm outputs it with probability >1/n2.
- Corollary: There are <n2 min cut-induced subsets of B

- Sampling according to connectivities gives a sparsifier
- We generalize Karger’s cut counting lemma

Questions

- Improve O(log2 n) to O(log n) in sampling analysis
- Applications of our cut-counting lemma?