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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) TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A A AAAAA A A A A

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Vertex sparsifiers: New results from old techniques (and some open questions) 2 Graph Bisection Input: Graph G=(V,E) Goal: partition the vertex set into V 1,V 2 with |V 1 |=|V 2 |, so as to minimize e(V 1,V 2 ). (may allow edge-capacities) Central problem, well-studied, NP-hard … Polynomial-time algorithm [Räcke08]: O(log n) approximation

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Vertex sparsifiers: New results from old techniques (and some open questions) 3 Terminal ( or Steiner ) Bisection Input: Graph G=(V,E) and terminals K µ V Goal: partition the vertex set into V 1,V 2 with |V 1 Å K|=| V 2 Å K |, so as to minimize e(V 1,V 2 ). Same O(log n) approximation [Räcke08]. But can we do f(k) where k=|K|? Similarly, Steiner versions of Linear Arrangement, Oblivious Routing, etc.

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Vertex sparsifiers: New results from old techniques (and some open questions) 4 Vertex Sparsifiers (w.r.t. Cuts) Input: Graph G=(V,E) and terminals K µ V Goal: A graph H on vertex set K, such that for every partition K=S [ T, MinCut G (S,T) ¼ MinCut H (S,T). (we allow edge-capacities) Why compress graph G onto terminal set K? Information-theory: Efficiently represent 2 k values Computation: Reduce problem size/approximation Stronger version: preserve all multi-commodity flows among terminals K. 5 9 4 8 3 5 2 1 9 8 G 4 3 1 H

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Vertex sparsifiers: New results from old techniques (and some open questions) 5 Vertex Sparsifiers – Previous work [Moitra09, Leighton-Moitra10] 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

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Vertex sparsifiers: New results from old techniques (and some open questions) 6 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 Convex combination of dominating trees Similar results (and lower bounds) were proved independently by Makarychev-Makarychev and by Charikar-Leighton-Li-Moitra.

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Vertex sparsifiers: New results from old techniques (and some open questions) 7 Best Previous Result Our ResultBest result for k=n Efficient Flow sparsifier- 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

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Vertex sparsifiers: New results from old techniques (and some open questions) 8 Flow–Distance Duality Connection between: Sparsifier: faithful representation of flows Embedding: faithful representation of distances Transfer Theorem [Räcke08, Andersen-Feige09]. 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 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. convex combination of mappings

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Vertex sparsifiers: New results from old techniques (and some open questions) 9 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 Z E £ E where M e,f = number of occurrences of f in M(e). Illustration: Embed V to a dominating tree T=(V,E T ) For xy 2 E T fix x-y path in G (e.g. shortest) Let M(uv 2 E) = {map u-v path in T into G}. But how to choose M ? G 1 n 2 3 s 13 e M(e) s 1n 2 3 n 1 T …

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Vertex sparsifiers: New results from old techniques (and some open questions) 10 0-extensions Defn: A 0-extension of (G=(V,E), l G ) with terminals K µ V to be a retraction f:V K; along with a graph (H=(K,E H ), l H ) where l H (x,y)=d G (x,y) for all (x,y) 2 E H. ) d H dominates d G [on pairs in K] G 1 n 2 3 2 3 n 1 H Defn: Stretch of a probabilistic 0-extension is the minimum ® ¸ 1 s.t. E H [d H (f(x),f(y))] · ® d G (x,y)for all x,y 2 V 30 10 20

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Vertex sparsifiers: New results from old techniques (and some open questions) 11 Tree 0-extensions Example 1: Graph H is a tree call it a tree 0-extension Corollary of [Gupta-Nagarajan-Ravi10]: There is an algorithm that produces tree 0-extensions with stretch ® =O(log k) Idea: Use variant of [Fakcharoenphol-Rao-Talwar04] but Allow distance between non-terminals to contract Remap non-terminals leaves to terminals Purge internal (Steiner) nodes [Gupta01] 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) G 1 n 2 3 2 3 n 1 H 30 10 20

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Vertex sparsifiers: New results from old techniques (and some open questions) 12 Induced 0-extensions Example 2: Graph H is induced by G via E H ={ (f(u),f(v)) : (u,v) 2 E } call this H=H f an induced 0-extension. Theorem [Fakcharoenphol-Harrelson-Rao-Talwar04]: 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) G 1 n 2 3 10 20 2 3 n 1 HfHf

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Vertex sparsifiers: New results from old techniques (and some open questions) 13 Planar Graphs Theorem [Calinescu-Karloff-Rabani04]: 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 H f is a minor of G. Hence planarity is guaranteed. We would like the sparsifier to be planar!!

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Vertex sparsifiers: New results from old techniques (and some open questions) 14 Connected 0-extension Defn. A 0-extension f:V K is called connected if each f -1 (x) induces a connected subgraph of G. Observe: f is connected ) H f is a minor of G ) H f 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 n 2 3 10 20 Connected: 1 n 2 3 10 20

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Vertex sparsifiers: New results from old techniques (and some open questions) 15 Implications to Metric Embedding Theorem [Gupta01]: For every tree T and terminals K, there is a tree on K that represents all distances faithfully (factor 8) 2 3 n 1 T … 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 2 T … 2 2 2

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Vertex sparsifiers: New results from old techniques (and some open questions) 16 Connected 0-extension in Planar Metrics Algorithm (Input: Graph G with edge-lengths l and terminals K) 1. Init: f(v)=v for v 2 K and f(v)= ? for v 2 V n K. 2. For each r=1,2,…,2 i,…,diam(V) 3. sample ¯ -decomposition P of d G with diameter r 4. for each C 2 P containing both mapped and unmapped vertices 5. delete from C mapped vertices 6. for each connected component C in C 7. choose vertex w C 2 C that was deleted and has edge to C 8. reset f(u)=f(w C ) for all u 2 C 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/r 2 (must be separate twiced) Pr[P(x) P(y)] · ¯ d G (x,y) / r

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Vertex sparsifiers: New results from old techniques (and some open questions) 17 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-Gupta08] 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)

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