1. Graph Sparsifiers by Edge-Connectivity and Random Spanning Trees Nick Harvey University of Waterloo Department of Combinatorics and Optimization Joint work with Isaac Fung TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A

2. What are sparsifiers? Approximating all cuts – Sparsifiers: number of edges = O(n log n / ² 2 ), 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 / ² 2 ), “entire spectrum” approximated within 1+ ². [SS’08] – O~(m) time algorithm to construct them [BSS’09] Poly(n) n = # vertices Laplacian matrix of G Laplacian matrix of Sparsifier Weighted subgraphs that approximately preserve some properties m = # edges Poly(n) [BSS’09]

3. Why are sparsifiers useful? Approximating all cuts – Sparsifiers: fast algorithms for cut/flow problem ProblemApproximationRuntimeReference Min st Cut1+ ² O~(n 2 )BK’96 Sparsest CutO(log n)O~(n 2 )BK’96 Max st Flow1O~(m+nv)KL’02 Sparsest CutO~(n 2 )AHK’05 Sparsest CutO(log 2 n)O~(m+n 3/2 )KRV’06 Sparsest CutO~(m+n 3/2+ ² )S’09 Perfect Matching in Regular Bip. Graphs n/aO~(n 1.5 )GKK’09 Sparsest CutO~(m+n 1+ ² )M’10 v = flow value n = # vertices m = # edges

4. Our Motivation BSS algorithm is very mysterious, and “too good to be true” Are there other methods to get sparsifiers with only O(n/ ² 2 ) edges? Wild Speculation: Union of O(1/ ² 2 ) 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(log 2 n/ ² 2 ) random spanning trees gives a sparsifier

5. Formal problem statement Design an algorithm such that Input: An undirected graph G=(V,E) Output: A weighted subgraph H=(V,F,w), where F µ E and w : F ! R Goals: | | ± G (U)| - w( ± H (U)) | · ² | ± G (U)| 8 U µ V (We only want to preserve cuts) |F| = O(n log n / ² 2 ) Running time = O~( m / ² 2 ) # edges between U and V\U in G weight of edges between U and V\U in H n = # vertices m = # edges | | ± (U)| - w( ± (U)) | · ² | ± (U)| 8 U µ V

6. Sparsifying Complete Graph Sampling: Construct H by sampling every edge of G with prob p=100 log n/n. Give each edge weight 1/p. Properties of H: # sampled edges = O(n log n) | ± G (U)| ¼ | ± H (U)| 8 U µ V So H is a sparsifier of G

7. Consider any cut ± G (U) with |U|=k. Then | ± G (U)| ¸ kn/2. Let X e = 1 if edge e is sampled. Let X =  e 2 C X e = | ± 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 Chernoff Bound Bound on # small cuts Key Ingredients Union bound Proof Sketch Exponentially increasing # of bad events Exponentially decreasing probability of failure

8. Generalize to arbitrary G? 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 Eliminate most of these

9. Non-uniform sampling algorithm [BK’96] Input: Graph G=(V,E), parameters p e 2 [0,1] Output: A weighted subgraph H=(V,F,w), where F µ E and w : F ! R For i=1 to ½ For each edge e 2 E With probability p e, Add e to F Increase w e by 1/( ½ p e ) Main Question: Can we choose ½ and p e ’s to achieve sparsification goals?

10. Non-uniform sampling algorithm [BK’96] Claim: H perfectly approximates G in expectation! For any e 2 E, E[ w e ] = 1 ) For every U µ V, E[ w( ± H (U)) ] = | ± G (U)| Goal: Show every w( ± H (U)) is tightly concentrated Input: Graph G=(V,E), parameters p e 2 [0,1] Output: A weighted subgraph H=(V,F,w), where F µ E and w : F ! R For i=1 to ½ For each edge e 2 E With probability p e, Add e to F Increase w e by 1/( ½ p e )

11. Prior Work Benczur-Karger ‘96 – Set ½ = O(log n), p e = 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 –  e p e · n ) |F| = O(n log n) (# edges in sparsifier) – Running time is O(m log 3 n) Spielman-Srivastava ‘08 – Set ½ = O(log n), p e = 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 –  e p e = n-1 ) |F| = O(n log n) (# edges in sparsifier) – Running time is O(m log 50 n) – Uses “Matrix Chernoff Bound” Assume ² is constant O(m log 3 n) [Koutis-Miller-Peng ’10] Similar to edge connectivity

12. Our Work Fung-Harvey ’10 (independently Hariharan-Panigrahi ‘10) – Set ½ = O(log 2 n), p e = 1/edge-connectivity of edge e – Edge-connectivity ¸ max { strength, effective conductance } –  e p e · n ) |F| = O(n log 2 n) – Running time is O(m log 2 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 Assume ² is constant Why? Pr[ e 2 T ] = effective resistance of e edges are negatively correlated ) Chernoff bound still works (min size of a cut that contains e)

13. Our Work Fung-Harvey ’10 (independently Hariharan-Panigrahi ‘10) – Set ½ = O(log 2 n), p e = 1/edge-connectivity of edge e – Edge-connectivity ¸ max { strength, effective conductance } –  e p e · n ) |F| = O(n log 2 n) – Running time is O(m log 2 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 log 2 n) + O~(n) Assume ² is constant (min size of a cut that contains e) O(n log n)

14. Our Work Fung-Harvey ’10 (independently Hariharan-Panigrahi ‘10) – Set ½ = O(log 2 n), p e = 1/edge-connectivity of edge e – Edge-connectivity ¸ max { strength, effective conductance } –  e p e · n ) |F| = O(n log 2 n) – Running time is O(m log 2 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 / ² 2 ) edges, with running time O(m) in unwtd graphs and O(m)+O~(n/ ² 2 ) in wtd graphs Assume ² is constant (min size of a cut that contains e)

15. Notation: k uv = min size of a cut separating u and v Main ideas: – Partition edges into connectivity classes E = E 1 [ E 2 [... E log n where E i = { e : 2 i-1 · k e <2 i } – Prove weight of sampled edges that each cut takes from each connectivity class is about right – Key point: Edges in ± (U) Å E i have nearly same weight – This yields a sparsifier U

16. Prove weight of sampled edges that each cut takes from each connectivity class is about right Notation: C = ± (U) is a cut C i = ± (U) Å E i is a cut-induced set Need to prove: C1C1 C2C2 C3C3 C4C4

17. Notation: C i = ± (U) Å E i is a cut-induced set C1C1 C2C2 C3C3 C4C4 Prove 8 cut-induced set C i Key Ingredients Chernoff bound: Prove small Bound on # small cuts: Prove #{ cut-induced sets C i induced by a small cut |C| } is small. Union bound: sum of failure probabilities is small, so probably no failures.

18. Counting Small Cut-Induced Sets Theorem: Let G=(V,E) be a graph. Fix any B µ E. Suppose k e ¸ K for all e in B. (k uv = min size of a cut separating u and v) Then, for every ® ¸ 1, |{ ± (U) Å B : | ± (U)| · ® K }| < n 2 ®. 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 }| < n 2 ®.

19. Comparison Theorem: Let G=(V,E) be a graph. Fix any B µ E. Suppose k e ¸ K for all e in B. (k uv = min size of a cut separating u and v) Then |{ ± (U) Å B : | ± (U)| · c }| < n 2c/K 8 c ¸ 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 }| < n 2c/K 8 c ¸ 1. How many cuts of size 1? Theorem says < n 2, taking K=c=1. Corollary, says < 1, because K=0. (Slightly unfair)

20. Comparison Theorem: Let G=(V,E) be a graph. Fix any B µ E. Suppose k e ¸ K for all e in B. (k uv = min size of a cut separating u and v) Then |{ ± (U) Å B : | ± (U)| · c }| < n 2c/K 8 c ¸ 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 }| < n 2c/K 8 c ¸ 1. Important point: A cut-induced set is a subset of edges. Many cuts can induce the same set. (Slightly unfair) ± (U’) ± (U)

21. 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/n 2. Corollary: There are · n 2 min cuts.

22. Finding a Small Cut-Induced Set Input: A graph G=(V,E), and B µ E 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 Claim: For any min cut-induced subset of B, this algorithm outputs it with probability > 1/n 2. Corollary: There are < n 2 min cut-induced subsets of B Splitting Off Replace edges {u,v} and {u’,v} with {u,u’} while preserving edge-connectivity Between all vertices other than v Splitting Off Replace edges {u,v} and {u’,v} with {u,u’} while preserving edge-connectivity Between all vertices other than v Wolfgang Mader v u u’ v u

23. Sparsifiers from Random Spanning Trees Let H be union of ½ =log 2 n uniform random spanning trees, where w e is 1/( ½ ¢ (effective resistance of e)) Then all cuts are preserved and |F| = O(n log 2 n) Why does this work? – Pr T [ e 2 T ] = effective resistance of edge e [Kirchoff 1847] – Similar to usual independent sampling algorithm, with p e = effective resistance of e – Key difference: edges in a random spanning tree are not independent, but they are negatively correlated! [BSST 1940] – Chernoff bounds still work. [Panconesi, Srinivasan 1997]

24. Sparsifiers from Random Spanning Trees Let H be union of ½ =log 2 n uniform random spanning trees, where w e is 1/( ½ ¢ (effective resistance of e)) Then all cuts are preserved and |F| = O(n log 2 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

25. Conclusions Graph sparsifiers important for fast algorithms and some combinatorial theorems Sampling by edge-connectivities gives a sparsifier with O(n log 2 n) edges in O(m log 2 n) time – Improvements: O(n log n) edges in O(m) + O~(n) time [Panigrahi ‘10] Sampling by effective resistances also works ) sampling O(log 2 n) random spanning trees gives a sparsifier Questions Improve log 2 n to log n? Sampling o(log n) random trees gives a sparsifier with o(log n) approximation?