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Graph Sparsifiers by Edge-Connectivity and Random Spanning Trees

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### Graph Sparsifiers byEdge-Connectivity andRandom Spanning Trees

Nick Harvey University of WaterlooDepartment of Combinatorics and Optimization

Joint work with Isaac Fung

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What are sparsifiers?

- Weighted subgraphs that approximately preserve some properties

[BSS’09]

- Approximating all cuts
- Sparsifiers: number of edges = O(n log n /²2) ,every cut approximated within 1+². [BK’96]
- O~(m) time algorithm to construct them
- Spectral approximation
- Spectral sparsifiers: number of edges = O(n log n /²2), “entire spectrum” approximated within 1+². [SS’08]
- O~(m) time algorithm to construct them

n = # vertices

Poly(n)

m = # edges

[BSS’09]

Laplacian matrix of G

Laplacian matrix of Sparsifier

Poly(n)

Why are sparsifiers useful?

- Approximating all cuts
- Sparsifiers: fast algorithms for cut/flow problem

v = flow value

Our Motivation

- BSS algorithm is very mysterious, and“too good to be true”
- Are there other methods to get sparsifierswith only O(n/²2) edges?
- Wild Speculation:Union of O(1/²2) random spanning trees gives a sparsifier(if weighted appropriately)
- True for complete graph [GRV ‘08]
- Corollary of our Main Result:The Wild Speculation is false, but the union of O(log2 n/²2)random spanning trees gives a sparsifier

Formal problem statement

- Design an algorithm such that
- Input: An undirected graph G=(V,E)
- Output: A weighted subgraphH=(V,F,w),where FµE and w : F !R
- Goals:
- | |±G(U)| - w(±H(U)) | ·² |±G(U)| 8U µ V

(We only want to preserve cuts)

- |F| = O(n log n / ²2)
- Running time = O~( m / ²2 )

# edges between U and V\U in G

weight of edges between U and V\U in H

- | |±(U)| - w(±(U)) | ·² |±(U)| 8U µ V

Sparsifying Complete Graph

- Sampling: Construct H by sampling every edge of Gwith probp=100 log n/n. Give each edge weight1/p.
- Properties of H:
- # sampled edges = O(n log n)
- |±G(U)| ¼ |±H(U)| 8U µ V
- So H is a sparsifier of G

Proof Sketch

- Consider any cut ±G(U) with |U|=k. Then |±G(U)|¸kn/2.
- Let Xe = 1 if edge e is sampled. Let X = e2CXe = |±H(U)|.
- Then ¹ = E[X] = p|±(U)| ¸ 50 k log n.
- Say cut fails if|X-¹| ¸¹/2.
- So Pr[ cut fails ] · 2 exp( - ¹/12 ) · n-4k.
- # of cuts with |U|=k is .
- So Pr[ any cut fails ] ·k n-4k < k n-3k < n-2.
- Whp, every U has ||±H(U)|-p|±(U)|| < p|±(U)|/2

Key Ingredients

Chernoff Bound

Bound on # small cuts

Union bound

Exponentially decreasingprobability of failure

Exponentially increasing # of bad events

Generalize to arbitrary G?

Eliminate most of these

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

Keep this

Non-uniform sampling algorithm [BK’96]

- Input: Graph G=(V,E), parameters pe2 [0,1]
- Output: A weighted subgraph H=(V,F,w),where FµE and w : F !R

- For i=1 to ½
- For each edge e2E
- With probability pe, Add e to F Increase we by 1/(½pe)

- Main Question: Can we choose ½ and pe’sto achieve sparsification goals?

Non-uniform sampling algorithm [BK’96]

- Input: Graph G=(V,E), parameters pe2 [0,1]
- Output: A weighted subgraph H=(V,F,w),where FµE and w : F !R

- For i=1 to ½
- For each edge e2E
- With probability pe, Add e to F Increase we by 1/(½pe)

- Claim: H perfectly approximates G in expectation!
- For any e2E, E[ we ] = 1
- ) For every UµV, E[ w(±H(U)) ] = |±G(U)|
- Goal: Show every w(±H(U)) is tightly concentrated

Prior Work

Similar to edge connectivity

- Benczur-Karger ‘96
- Set ½ = O(log n), pe = 1/“strength” of edge e(max k s.t. e is contained in a k-edge-connected vertex-induced subgraph of G)
- All cuts are preserved
- epe·n ) |F| = O(n log n) (# edges in sparsifier)
- Running time is O(m log3 n)
- Spielman-Srivastava ‘08
- Set ½ = O(log n), pe = 1/“effective conductance” of edge e(view G as an electrical network where each edge is a 1-ohm resistor)
- H is a spectral sparsifier of G ) all cuts are preserved
- epe=n-1 ) |F| = O(n log n) (# edges in sparsifier)
- Running time is O(m log50 n)
- Uses “Matrix Chernoff Bound”

O(m log3 n)[Koutis-Miller-Peng ’10]

Our Work

- Fung-Harvey ’10 (independentlyHariharan-Panigrahi ‘10)
- Set ½ = O(log2 n), pe = 1/edge-connectivity of edge e
- Edge-connectivity¸ max { strength, effective conductance }
- epe·n ) |F| = O(n log2 n)
- Running time is O(m log2 n)
- Advantages:
- Edge connectivities natural, easy to compute
- Faster than previous algorithms
- Implies sampling by edge strength, effective resistances,or random spanning trees works
- Disadvantages:
- Extra log factor, no spectral sparsification

(min size of a cut that contains e)

Why?

Pr[ e 2 T ] = effective resistance of eedges are negatively correlated

)Chernoff bound still works

Our Work

- Fung-Harvey ’10 (independentlyHariharan-Panigrahi ‘10)
- Set ½ = O(log2 n), pe = 1/edge-connectivity of edge e
- Edge-connectivity¸ max { strength, effective conductance }
- epe·n ) |F| = O(n log2 n)
- Running time is O(m log2 n)
- Advantages:
- Edge connectivities natural, easy to compute
- Faster than previous algorithms
- Implies sampling by edge strength, effective resistances…
- Extra trick:Can shrink |F| to O(n log n) by using Benczur-Karger to sparsify our sparsifier!
- Running time is O(m log2 n) + O~(n)

(min size of a cut that contains e)

O(n log n)

Our Work

- Fung-Harvey ’10 (independentlyHariharan-Panigrahi ‘10)
- Set ½ = O(log2 n), pe = 1/edge-connectivity of edge e
- Edge-connectivity¸ max { strength, effective conductance }
- epe·n ) |F| = O(n log2 n)
- Running time is O(m log2 n)
- Advantages:
- Edge connectivities natural, easy to compute
- Faster than previous algorithms
- Implies sampling by edge strength, effective resistances…
- Panigrahi ’10
- A sparsifier with O(n log n /²2) edges, with running timeO(m) in unwtd graphs and O(m)+O~(n/²2) in wtd graphs

(min size of a cut that contains e)

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
- Key point: Edges in ±(U)ÅEi have nearly same weight
- This yields a sparsifier

U

- C = ±(U) is a cut
- Ci= ±(U) ÅEi is a cut-induced set
- Need to prove:

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

C2

C3

C1

C4

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

Prove 8 cut-induced set Ci

- Key Ingredients
- Chernoff bound: Provesmall
- Bound on # small cuts: Prove #{ cut-induced sets Ciinduced by a small cut |C| }is small.
- Union bound: sum of failure probabilities is small,so probably no failures.

C2

C3

C1

C4

Counting Small Cut-Induced Sets

- Theorem: 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®.

- Corollary: Counting Small Cuts[K’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®.

Comparison

(Slightly unfair)

- Theorem: 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 |{ ±(U) ÅB : |±(U)|·c }| < n2c/K8c¸1.

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

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

Then, |{ ±(U) : |±(U)|·c }| < n2c/K 8c¸1.

- How many cuts of size 1?

Theorem says < n2, taking K=c=1.

Corollary, says < 1, because K=0.

Comparison

(Slightly unfair)

- Theorem: 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 |{ ±(U) ÅB : |±(U)|·c }| < n2c/K8c¸1.

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

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

Then, |{ ±(U) : |±(U)|·c }| < n2c/K 8c¸1.

- Important point: A cut-induced set is a subset of edges. Many cuts can induce the same set.

±(U’)

±(U)

Algorithm for Finding a Min Cut [K’93]

- 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.

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

Between all vertices other than v

Finding a Small Cut-Induced Setv

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
- 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

Sparsifiers from Random Spanning Trees

- Let H be union of ½=log2 n uniform random spanning trees,where we is 1/(½¢(effective resistance of e))
- Then all cuts are preserved and |F| = O(n log2 n)
- Why does this work?
- PrT[ e 2 T ] = effective resistance of edge e [Kirchoff 1847]
- Similar to usual independent sampling algorithm,with pe = effective resistance of e
- Key difference: edges in a random spanning tree arenot independent, but they are negatively correlated![BSST 1940]
- Chernoff bounds still work. [Panconesi, Srinivasan 1997]

Sparsifiers from Random Spanning Trees

- Let H be union of ½=log2 n uniform random spanning trees,where we is 1/(½¢(effective resistance of e))
- Then all cuts are preserved and |F| = O(n log2 n)
- How is this different than independent sampling?
- Consider an n-cycle. There are n/2 disjoint cuts of size 2.
- When ½=1, each cut has constant prob of having no edges

) need ½=(log n) to get a connected graph

- With random trees, get connectivity after just one tree
- Are O(1) trees are enough to preserve all cuts?
- No! ( log n ) trees are required

Conclusions

- Graph sparsifiers important for fast algorithms and some combinatorial theorems
- Sampling by edge-connectivities gives a sparsifierwith O(n log2 n) edges in O(m log2 n) time
- Improvements: O(n log n) edges in O(m) + O~(n) time[Panigrahi ‘10]
- Sampling by effective resistances also works

) sampling O(log2 n) random spanning trees gives a sparsifier

Questions

- Improve log2 n to log n?
- Sampling o(log n) random trees gives a sparsifier with o(log n) approximation?

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