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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)
Vertex sparsifiers: New results from old techniques (and some open questions)
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Graph BisectionInput: 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) BisectionInput: 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)Input: Graph G=(V,E) and terminals KµV
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)
Why “compress” graph G “onto” terminal set K? Information-theory: Efficiently represent 2k values Computation: Reduce problem size/approximation
Stronger version: preserve all multi-commodity flows among terminals K.
5 94
8
35
2
1
9
8
G
4
31
H
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 .
Can efficiently find one with quality .
Yields approximation for Terminal Bisection
Similarly, approximation for other problems
But H is not “simple” Even if G is
O( logklog logk )
O( log2 klog logk )
O( log3 klog logk )
polylog(k)
Vertex sparsifiers: New results from old techniques (and some open questions)
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Our Results Can efficiently find a sparsifier with quality .
Can efficiently find a tree-based sparsifier with quality .
Yields approximation for Terminal Bisection.
Similar improvements for other problems
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
O( logklog logk )
O(logk)
O(logk)
Convex combination of “dominating” trees
Similar results (and lower bounds) were proved independently by Makarychev-Makarychev and by Charikar-Leighton-Li-Moitra.
Vertex sparsifiers: New results from old techniques (and some open questions)
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Vertex sparsifiers: New results from old techniques (and some open questions)
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Flow–Distance DualityConnection 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:EP(E). Then: For all edge-lengths l:ER+ there is a probabilistic mapping with
stretch (distortion) ½¸1 m
For all edge-capacities c:ER+ there is a probabilistic mapping with quality (congestion) ½¸1
Moreover, there is efficient algorithm for one iff for the other.
convex combination of mappings
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:EP(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}.
But how to choose M?
G1
n23
s13
e
M(e)s1n
2 3 n
1T
…
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]
G1
n23 2 3 n
1H
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
3010
20
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
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 Allow distance between non-terminals to contract “Remap” non-terminals leaves to terminals “Purge” internal (Steiner) nodes [Gupta’01]
Now use the Transfer Theorem: M = all tree 0-extensions Distance mappings exist with stretch O(log k) Thus get a tree-based sparsifier with quality O(log k)
G1
n23 2 3 n
1H
3010
20
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.
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)
G1
n23
10
20
2 3 n
1Hf
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)
Idea: Make sure Hf is a minor of G. Hence planarity is guaranteed.
We would like the sparsifier to be planar!!
Vertex sparsifiers: New results from old techniques (and some open questions)
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Connected 0-extension Defn. A 0-extension f:VK is called connected if each f-1(x) induces
a connected subgraph of G.
Observe: f is connected ) Hf is a minor of G ) Hf is planar
We give first algorithms for connected 0-extension: 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.
Not connected:1
n23
10
20Connected:
1
n23
10
20
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)
2 3 n
1T
…
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
3 4 n
2T’
…
22 2
Vertex sparsifiers: New results from old techniques (and some open questions)
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Connected 0-extension in Planar MetricsAlgorithm (Input: Graph G with edge-lengths l and terminals K)1. Init: f(v)=v for v2K and f(v)=? for v2VnK.
2. For each r=1,2,…,2i,…,diam(V)
3. sample ¯-decomposition P of dG with diameter r
4. for each C’2P containing both mapped and unmapped vertices
5. delete from C’ mapped vertices
6. for each connected component C in C’
7. choose vertex wC2C’ that was deleted and has edge to C
8. 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)
Pr[P(x)P(y)] · ¯ dG(x,y) / r
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 Apparently possible for outerplanar graphs [Basu-Gupta’08] More generally: same for general G, using minors
s-sparse extension: Given a graph G and terminals K, choose S¶K of size s, and a 0-extension (retraction) into this S Is there a poly(k)-sparse extension of expected stretch O(1)?
Is there a single (non-probabilistic) planar sparsifier graph? 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”)