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### Graph Sparsifiers: A Survey

Based on work by: Batson, Benczur, de Carli Silva, Fung, Hariharan, Harvey, Karger, Panigrahi, Sato, Spielman, Srivastava and Teng

Nick Harvey

Approximating Dense Objects by Sparse Ones

- Floor joists
- Image compression

Approximating Dense Graphsby Sparse Ones

- Spanners: Approximate distances to within ®using only = O(n1+2/®) edges
- Low-stretch trees: Approximate most distancesto within O(log n) using only n-1 edges

(n = # vertices)

Overview

- Definitions
- Cut & Spectral Sparsifiers
- Cut Sparsifiers
- A combinatorial construction
- Spectral Sparsifiers
- A random sampling construction
- Derandomization

Cut Sparsifiers

- Input: An undirected graph G=(V,E) with weights u : E !R+
- Output: A subgraph H=(V,F) of G with weightsw : F!R+ such that |F| is small and

w(±H(U)) = (1§²) u(±G(U)) 8Uµ V

weight of edges between U and V\U in H

weight of edges between U and V\U in G

U

U

Cut Sparsifiers

- Input: An undirected graph G=(V,E) with weights u : E !R+
- Output: A subgraph H=(V,F) of G with weightsw : F!R such that |F| is small and

w(±H(U)) = (1§²) u(±G(U)) 8Uµ V

weight of edges between U and V\U in H

weight of edges between U and V\U in G

Generic Applicationof Cut Sparsifiers

(Slow) Algorithm A for some problem P

(Dense) Input graph G

Exact/Approx Output

Min s-t cut, Sparsest cut,Max cut, …

(Efficient) Sparsification Algorithm S

Algorithm A(now faster)

Sparse graph H approx preserving solution of P

Approximate Output

Relation to Expander Graphs

- Graph H on V is an expander if, for some constant c,|±H(U)| ¸c¢ |U| 8Uµ V, |U|·n/2
- Let G be the complete graph on V. Note that|±G(U)| = |U|¢|VnU| · n¢|U|
- If we give all edges of H weight w=n, thenw(±H(U)) ¸ c¢|±G(U)| 8Uµ V, |U|·n/2
- Expanders are similar to sparsifiers of complete graph

G

H

Relation to Expander Graphs

- Fact: Pick a random graph where each edge appears independently with probability p=(log(n)/n). Gives an expander with O(n log n) edges with high probability.

G

H

Spectral Sparsifiers

- Input: An undirected graph G=(V,E) withweights u : E !R+
- Def: The Laplacianis the matrix LG such thatxTLGx = st2Eust (xs-xt)28x2RV.
- LG is positive semidefinite since this is ¸ 0.
- Example: Electrical Networks
- View edge st as resistor of resistance 1/ust.
- Impose voltage xv at every vertex v.
- Ohm’s Power Law: P = V2/R.
- Power consumed on edge st is ust (xs-xt)2.
- Total power consumed is xTLGx.

Spectral Sparsifiers

- Input: An undirected graph G=(V,E) withweights u : E !R+
- Def: The Laplacianis the matrix LG such thatxTLGx = st2Eust (xs-xt)28x2RV.
- Output: A subgraphH=(V,F) of G with weightsw : F!R such that |F| is small and

xTLHx = (1 §²) xTLGx8x2RV

w(±H(U)) = (1 § ²) u(±G(U)) 8Uµ V

SpectralSparsifier

)

)

CutSparsifier

Cut vs Spectral Sparsifiers

- Number of Constraints:
- Cut: w(±H(U)) = (1§²) u(±G(U)) 8UµV (2n constraints)
- Spectral: xTLHx = (1§²) xTLGx8x2RV (1 constraints)
- Spectral constraints are SDP feasibility constraints:

(1-²) xTLGx· xTLHx· (1+²) xTLGx8x2RV

, (1-²) LG¹LH¹ (1+²) LG

- Spectral constraints are actually easier to handle
- Checking “Is H is a spectral sparsifier of G?” is in P
- Checking “Is H is a cut sparsifier of G?” isnon-uniform sparsest cut, so NP-hard

Here X ¹ Y means Y-X is positive semidefinite

Application of Spectral Sparsifiers

- Consider the linear system LGx = b.Actual solution is x := LG-1 b.
- Instead, compute y := LH-1b,where H is a spectral sparsifier of G.
- We know: (1-²) LG¹LH¹(1+²) LG

)y has low multiplicative error: ky-xkLG·2²kxkLG

- Computing y is fast since H is sparse:conjugate gradient method takes O(n|F|) time (where |F| = # nonzero entries of LH)

Application of Spectral Sparsifiers

- Consider the linear system LGx = b.Actual solution is x := LG-1 b.
- Instead, compute y := LH-1b,where H is a spectral sparsifier of G.
- We know: (1-²) LG¹LH¹(1+²) LG

)y has low multiplicative error: ky-xkLG·2²kxkLG

- Theorem:[Spielman-Teng ‘04, Koutis-Miller-Peng ‘10]Can compute avector y with low multiplicative error in O(m log n (log log n)2) time. (m = # edges of G)

Results on Sparsifiers

Cut Sparsifiers

Spectral Sparsifiers

Karger ‘94

Benczur-Karger ‘96

Combinatorial

Fung-Hariharan-Harvey-Panigrahi ‘11

Spielman-Teng ‘04

Spielman-Srivastava ‘08

Linear Algebraic

Batson-Spielman-Srivastava ‘09

de Carli Silva-Harvey-Sato ‘11

These construct sparsifiers with n logO(1) n / ²2 edges

These construct sparsifiers with O(n / ²2) edges

Sparsifiers by Random Sampling

- The complete graph is easy!Random sampling gives an expander (ie. sparsifier) with O(n log n) edges.

Sparsifiers by Random Sampling

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

- Choose parameter ½
- Compute probabilities { pe:e2E}
- For i=1 to ½
- For each edge e2E
- With probability pe, Add e to F Increase we by ue/(½pe)

Can we dothis so that thecut values aretightly concentratedand E[|F|]=nlogO(1)n?

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

Benczur-Karger ‘96

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

- Choose parameter ½
- Compute probabilities { pe:e2E}
- For i=1 to ½
- For each edge e2E
- With probability pe, Add e to F Increase we by ue/(½pe)

Can we dothis so that thecut values aretightly concentratedand E[|F|]=nlogO(1)n?

Can approximateall values in

m logO(1)n time.

- Set ½ = O(logn/²2).
- Let pe = 1/“strength” of edge e.
- Cuts are preserved to within (1§²) and E[|F|] = O(n logn/²2)

Fung-Hariharan-Harvey-Panigrahi ‘11

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

- Choose parameter ½
- Compute probabilities { pe:e2E}
- For i=1 to ½
- For each edge e2E
- With probability pe, Add e to F Increase we by ue/(½pe)

Can we dothis so that thecut values aretightly concentratedand E[|F|]=nlogO(1)n?

Can approximateall values in

O(m + n log n) time

- Set ½ = O(log2n/²2).
- Let pst = 1/(min cut separating s and t)
- Cuts are preserved to within (1§²) and E[|F|] = O(n log2n/²2)

Letkuv = min size of a cut separating u and v.Recall sampling probability is pe = 1/ke

- 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 has low error
- Key point: Edges in ±(U)ÅEi have roughly same pe
- This yields a sparsifier

U

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

Prove weight of sampled edges that each cuttakes from each connectivity class has low error

C2

C3

C1

C4

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

Prove 8 cut-induced set Ci

- Key Ingredients
- Hoeffding bound: Provesmall
- Bound on # small cut-induced sets:For most of these events, u(C) is large.In other words, #{ cut-induced sets Ciinduced by a small cut C }is small.

C2

C3

C1

C4

Counting Small Cut-Induced Sets

- Theorem:[Fung-Hariharan-Harvey-Panigrahi ‘11]

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: [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®.

Summary for Cut Sparsifiers

- Do non-uniform sampling of edges,with probabilities based on “connectivity”
- Analysis involves:
- Decomposing the graph
- Hoeffding bounds to analyze each “cut”
- Cut-counting theorem: “few small cuts”
- BK’96 had weaker cut-counting theorem, but had more complicated “connectivity” notion.
- Can get sparsifiers with O(n log n / ²2) edges
- Optimal for any independent sampling algorithm

Spectral Sparsification

- Input: Graph G=(V,E), weights u : E !R+
- Recall: xTLGx = st2Eust (xs-xt)2
- Goal: Find weights w : E !R+ such that very fewwe are non-zero, and(1-²) xTLGx·e2EwexTLex·(1+²) xTLGx8x2RV

, (1- ²) LG¹e2EweLe¹ (1+²) LG

- General Problem: Given matrices Le satisfying eLe=LG, find coefficients we, mostly zero, such that (1-²) LG¹eweLe¹ (1+²) LG

Call this xTLstx

The General Problem:Sparsifying Sums of PSD Matrices

- General Problem: Given PSD matrices Les.t. eLe=LG, find coefficients we, mostly zero, such that (1-²) LG¹eweLe¹ (1+²) LG
- Theorem:[Ahlswede-Winter ’02]Randomized alg gives w with O( n log n/²2 ) non-zeros.
- Theorem:[de Carli Silva-Harvey-Sato ‘11],building on [Batson-Spielman-Srivastava ‘09]Deterministic alg gives w with O( n/²2 ) non-zeros.
- Cut & spectral sparsifiers with O(n/²2) edges [BSS’09]
- Sparsifiers with more properties and O(n/²2) edges [dHS’11]

Vector Case

- Vector problem: Given vectors ve2[0,1]ns.t. eve = v,find coefficients we, mostly zero, such thatkeweve - v k1 · ²
- Theorem [Althofer ‘94, Lipton-Young ‘94]:There is a w with O(log n/²2) non-zeros.
- Proof: Random sampling & Hoeffding inequality.
- Multiplicative version: There is a w with O(n log n/²2) non-zeros such that (1-²) v·eweve· (1+²) v

- General Problem: Given PSD matrices Les.t. eLe = L, find coefficients we, mostly zero, such that (1-²) L¹eweLe¹ (1+²) L

Concentration Inequalities

- Theorem: [Chernoff ‘52, Hoeffding ‘63]Let Y1,…,Yk be i.i.d. randomnon-negative real numberss.t. E[ Yi ] = Z andYi·uZ. Then
- Theorem: [Ahlswede-Winter ‘02]Let Y1,…,Yk be i.i.d. randomPSD nxn matricess.t. E[ Yi ] = Z andYi¹uZ. Then

The only difference

“Balls & Bins” Example

- Problem: Throw k balls into n bins. Wantmax load / min load · 1+². How big should k be?
- AW Theorem: Let Y1,…,Yk be i.i.d. random PSD matricessuch that E[ Yi ] = Z andYi¹uZ. Then
- Solution: Let Yi be all zeros, except for a single n in a random diagonal entry.

Then E[ Yi ] = I =: Z, and ¸max(Yi Z-1) = n =: u.Set k = £(n logn/²2). Then, with high probability,every diagonal entry of i Yi/k is in [1-²,1+²].

Solving the General Problem

- General Problem: Given PSD matrices Les.t. eLe = LG, find coefficients we, mostly zero, such that (1-²) LG¹eweLe¹ (1+²) LG
- AW Theorem: Let Y1,…,Yk be i.i.d. random PSD matricessuch that E[ Yi ] = Z andYi¹uZ. Then
- Solve General Problem with O(nlogn/²2) non-zeros
- Repeat k:=£(nlogn/²2) times
- Pick an edge e with probability pe := Tr(LeLG-1)/n
- Increment we by 1/k¢pe

Derandomization

- Vector problem: Given vectors ve2[0,1]ns.t. eve = v,find coefficients we, mostly zero, such thatkeweve - vk1 · ²
- Theorem[Young ‘94]: The multiplicative weights method deterministically gives w with O(log n/²2) non-zeros
- Or, use pessimistic estimators on the Hoeffding proof
- General Problem: Given PSD matrices Les.t. eLe = LG, find coefficients we, mostly zero, such that (1-²) LG¹eweLe¹ (1+²) LG
- Theorem [de Carli Silva-Harvey-Sato ‘11]:The matrix multiplicative weights method (Arora-Kale ‘07)deterministically gives w with O(n log n/²2) non-zeros
- Or, use matrix pessimistic estimators (Wigderson-Xiao ‘06)

MWUM for “Balls & Bins”

- Let ¸i = load in bin i. Initially ¸=0. Want: 1·¸i and ¸i·1.
- Introduce penalty functions “exp(l-¸i)” and “exp(¸i-u)”
- Find a bin ¸i to throw a ball into such that,increasing l by ±l and u by ±u, the penalties don’t grow.

i exp(l+±l - ¸i’) · i exp(l-¸i)

i exp(¸i’-(u+±u)) · i exp(¸i-u)

- Careful analysis shows O(n log n/²2) balls is enough

¸ values:

u

l

0

1

MMWUM for General Problem

- Let A=0 and ¸ its eigenvalues. Want: 1·¸i and ¸i·1.
- Use penalty functions Tr exp(lI-A) and Tr exp(A-uI)
- Find a matrix Le such that adding ®Le to A,increasing l by ±l and u by ±u, the penalties don’t grow.

Trexp((l+±l)I- (A+®Le)) · Trexp(lI-A)

Trexp((A+®Le)-(u+±u)I) · Tr exp(A-uI)

- Careful analysis shows O(n log n/²2) matrices is enough

¸ values:

u

l

0

1

Beating Sampling & MMWUM

- To get a better bound, try changing the penalty functions to be steeper!
- Use penalty functions Tr (A-lI)-1 and Tr (uI-A)-1
- Find a matrix Le such that adding ®Le to A,increasing l by ±l and u by ±u, the penalties don’t grow.

Tr((A+®Le)-(l+±l)I)-1· Tr(A-lI)-1

Tr((u+±u)I-(A+®Le))-1· Tr (uI-A)-1

All eigenvaluesstay within [l, u]

¸ values:

u

l

0

1

Beating Sampling & MMWUM

- To get a better bound, try changing the penalty functions to be steeper!
- Use penalty functions Tr (A-lI)-1 and Tr (uI-A)-1
- Find a matrix Le such that adding ®Le to A,increasing l by ±l and u by ±u, the penalties don’t grow.

Tr((A+®Le)-(l+±l)I)-1· Tr(A-lI)-1

Tr((u+±u)I-(A+®Le))-1· Tr (uI-A)-1

- General Problem: Given PSD matrices Les.t. eLe = LG, find coefficients we, mostly zero, such that (1-²) LG¹eweLe¹ (1+²) LG
- Theorem:[Batson-Spielman-Srivastava ‘09] in rank-1 case,[de Carli Silva-Harvey-Sato ‘11] for general caseThis gives a solution w with O( n/²2 ) non-zeros.

Applications

- Theorem:[de Carli Silva-Harvey-Sato ‘11]Given PSD matrices Les.t. eLe = L, there is analgorithm to find w with O( n/²2 ) non-zeros such that (1-²) L¹eweLe¹ (1+²) L
- Application 1: Spectral Sparsifiers with CostsGiven costs on edges of G, can find sparsifier H whose cost isat most (1+²) the cost of G.
- Application 2: Simultaneous Spectral SparsifiersGiven two graphs G1 & G2 with a bijection on their edges,can choose edges that simultaneously sparsify G1 & G2.
- Application 3: Sparse SDP Solutionsmin { cTy : iyiAiº B, y¸0 } where Ai’s and B are PSDhas nearly optimal solution with O(n/²2) non-zeros.

Open Questions

- Use of sparsifiers in other areas (infoviz, etc.)
- Sparsifiers for directed graphs
- Construction of expander graphs
- More control of the weights we
- A combinatorial proof of spectral sparsifiers
- More applications of our general theorem

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