Vertex sparsifiers: New results
from old techniques
(and some open questions)
Robert Krauthgamer (Weizmann Institute)
Joint work with Matthias Englert, Anupam Gupta,
Harald Räcke, Inbal Talgam-Cohen and Kunal Talwar.
Presented: Newton Institute, Jan. 2011 (w/minor corrections)
Graph Bisection
Input: Graph G=(V,E)
Goal: partition the vertex set into V1,V2
with |V1|=|V2|,
so as to minimize e(V1,V2).
(may allow edge-capacities)


Central problem, well-studied, NP-hard …
Polynomial-time algorithm [Räcke’08]:
O(log n) approximation
Vertex sparsifiers: New results from old techniques (and some open questions)
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Terminal (or Steiner) Bisection
Input: Graph G=(V,E) and terminals KµV
Goal: partition the vertex set into V1,V2
with |V1ÅK|=| V2ÅK |,
so as to minimize e(V1,V2).

Same O(log n) approximation [Räcke’08].

But can we do f(k) where k=|K|?
Similarly, Steiner versions of Linear Arrangement, Oblivious Routing, etc.
Vertex sparsifiers: New results from old techniques (and some open questions)
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Vertex Sparsifiers (w.r.t. Cuts)
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Input: Graph G=(V,E) and terminals KµV
8
1
G
Goal: A graph H on vertex set K, such that
for every partition K=S[T,
MinCutG(S,T) ¼ MinCutH(S,T).
(we allow edge-capacities)
9
3
5
8
4

5
9
Why “compress” graph G “onto” terminal set K?


Information-theory: Efficiently represent 2k values
Computation: Reduce problem size/approximation
H
1
3

Stronger version: preserve all multi-commodity
flows among terminals K.
4
Vertex sparsifiers: New results from old techniques (and some open questions)
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Vertex Sparsifiers – Previous work
[Moitra’09, Leighton-Moitra’10]



There are (flow) sparsifiers with quality O( log k ) .
log l og k
Can efficiently find one with quality

l og3 k
Yields O(
l og l og k

Similarly,
l og2 k
O( l og l og k
).
) approximation for Terminal Bisection
polylog(k)
approximation for other problems
But H is not “simple”

Even if G is
Vertex sparsifiers: New results from old techniques (and some open questions)
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Our Results

log k
Can efficiently find a sparsifier with quality O( log l og k ).
Convex combination of
“dominating” trees

Can efficiently find a tree-based sparsifier with quality O(logk).

Yields

O(logk) approximation for Terminal Bisection.
Similar improvements for other problems
Similar results (and lower bounds) were proved independently by
Makarychev-Makarychev and by Charikar-Leighton-Li-Moitra.

If G is planar, then quality is O(1) and H is planar-based


In fact, only use minors of G
Holds for every minor-closed family
Vertex sparsifiers: New results from old techniques (and some open questions)
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Best Previous
Result
Our Result
Efficient Flow sparsifier
Best result for
k=n
-
Tree-based flow sparsifier
Minor-based flow sparsifier
-
-
Steiner Oblivious Routing
Steiner Min Lin. Arr.
Steiner MLA in planar graphs
Steiner Min cut Lin. Arr.
Steiner Bisection
Vertex sparsifiers: New results from old techniques (and some open questions)
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Flow–Distance Duality
Connection between:
 Sparsifier: faithful representation of flows
 Embedding: faithful representation of distances
Transfer Theorem [Räcke’08, Andersen-Feige’09].
Fix a graph G and a collection M of mappings M:EP(E). Then:
 For all edge-lengths l:ER+ there is a probabilistic mapping with
stretch (distortion) ½¸1
convex combination of mappings
m
 For all edge-capacities c:ER+ there is a probabilistic mapping with
quality (congestion) ½¸1
Moreover, there is efficient algorithm for one iff for the other.
Vertex sparsifiers: New results from old techniques (and some open questions)
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Edge Mappings


Fix G=(V,E), and let P(E) be all multisets of E (typically paths).
A mapping M:EP(E) can be represented as a matrix M in ZE£E
where Me,f = number of occurrences of f in M(e).
Illustration:
 Embed V to a dominating tree T=(V,ET)
 For xy2ET fix x-y path in G (e.g. shortest)
 Let M(uv2E) = {“map” u-v path in T into G}.
G
1
s13
M(e)
2
s1n
n
e 3
T
1
2 3
…
n
But how to choose M?
Vertex sparsifiers: New results from old techniques (and some open questions)
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0-extensions

Defn: A 0-extension of (G=(V,E), lG) with terminals KµV to be


a retraction f:V K;
along with a graph (H=(K,EH),lH) where lH(x,y)=dG(x,y) for all (x,y)2 EH.
) dH dominates dG [on pairs in K]
H
G
1
2

1
20
30
10
3
n
2 3
n
Defn: Stretch of a probabilistic 0-extension is the minimum ®¸1 s.t.
EH[dH(f(x),f(y))] · ® dG(x,y)
for all x,y2V
Vertex sparsifiers: New results from old techniques (and some open questions)
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Tree 0-extensions

Example 1: Graph H is a tree  call it a tree 0-extension
G
1
2


1
10
3
30
n
2 3
n
Corollary of [Gupta-Nagarajan-Ravi’10]: There is an algorithm that
produces tree 0-extensions with stretch ®=O(log k)
Idea: Use variant of [Fakcharoenphol-Rao-Talwar’04] but

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
H
20
Allow distance between non-terminals to contract
“Remap” non-terminals leaves to terminals
“Purge” internal (Steiner) nodes [Gupta’01]
Now use the Transfer Theorem:

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
M = all tree 0-extensions
Distance mappings exist with stretch O(log k)
Thus get a tree-based sparsifier with quality O(log k)
Vertex sparsifiers: New results from old techniques (and some open questions)
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Induced 0-extensions

Example 2: Graph H is “induced by G” via EH={ (f(u),f(v)) : (u,v)2 E }
 call this H=Hf an induced 0-extension.
G
1
2
Hf
20
1
10
3
n
2
3
n

Theorem [Fakcharoenphol-Harrelson-Rao-Talwar’04]: There is an
algorithm producing induced 0-extensions with ®=O(log k / loglog k)

Now use the Transfer Theorem:



M = all induced 0-extensions
Distance mappings exist with stretch O(log k / loglog k)
Thus get a sparsifier with quality O(log k / loglog k)
Vertex sparsifiers: New results from old techniques (and some open questions)
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Planar Graphs

Theorem [Calinescu-Karloff-Rabani’04]: There is an algorithm
producing induced 0-extensions with ®=O(1)

Now use the Transfer Theorem:



M = all induced 0-extensions
Distance mappings exist with stretch O(1)
Thus get a sparsifier with quality O(1)
We would like the sparsifier to be planar!!

Idea: Make sure Hf is a minor of G. Hence planarity is guaranteed.
Vertex sparsifiers: New results from old techniques (and some open questions)
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Connected 0-extension

Defn. A 0-extension f:VK is called connected if each f-1(x) induces
a connected subgraph of G.
Connected:
1
2
Not connected:
20
1
10
3
n
2
20
10
3

Observe: f is connected ) Hf is a minor of G ) Hf is planar
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We give first algorithms for connected 0-extension:
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n
For planar graphs: we achieve stretch O(1)
For ¯-decomposable metrics: stretch O(¯ log ¯)
For general metrics: stretch O(log k)
Via Transfer Theorem: planar-based sparsifier with quality O(1) etc.
Vertex sparsifiers: New results from old techniques (and some open questions)
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Implications to Metric Embedding

Theorem [Gupta’01]: For every tree T and terminals K, there is a
tree on K that represents all distances faithfully (factor 8)
T
T’
1
2
2
2 3


…
n
3 4
2
2
…
n
This work: For every planar graph G and terminals K, there is a
(probabilistic) planar graph on K that represents all distances
faithfully (expected O(1) stretch)
Simplifies embedding results
Vertex sparsifiers: New results from old techniques (and some open questions)
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Connected 0-extension in Planar Metrics
Algorithm (Input: Graph G with edge-lengths l and terminals K)
1.
2.
3.
4.
5.
6.
7.
8.
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Init: f(v)=v for v2K and f(v)=? for v2VnK.
Pr[P(x)P(y)] · ¯ dG(x,y) / r
For each r=1,2,…,2i,…,diam(V)
sample ¯-decomposition P of dG with diameter r
for each C’2P containing both mapped and unmapped vertices
delete from C’ mapped vertices
for each connected component C in C’
choose vertex wC2C’ that was deleted and has edge to C
reset f(u)=f(wC) for all u2C
G
Connectivity: by construction
Diameter: at time r, vertices are
mapped to terminals within O(r)
Stretch: Prob. to settle (u,v) at “late”
time r is 1/r2 (must be separate twiced)
Vertex sparsifiers: New results from old techniques (and some open questions)
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Open Problems

Steiner Points Removal: Given planar graph G and terminals K,
build a single planar graph only on K that represents all distances
faithfully
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s-sparse extension: Given a graph G and terminals K, choose S¶K
of size s, and a 0-extension (retraction) into this S

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Is there a poly(k)-sparse extension of expected stretch O(1)?
Is there a single (non-probabilistic) planar sparsifier graph?

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Apparently possible for outerplanar graphs [Basu-Gupta’08]
More generally: same for general G, using minors
More generally: extend duality between Distances and Capacities,
perhaps to level of a single graph, or to “preserve” minors
Analogous questions for cuts (e.g. SPR, few “pseudo-terminals”)
Analogous questions for Euclidean metrics (e.g. what is “minor”)
Vertex sparsifiers: New results from old techniques (and some open questions)
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