Graph Sparsifiers by Edge-Connectivity and Random Spanning Trees

Graph Sparsifiers byEdge-Connectivity andRandom Spanning Trees Nick HarveyU. Waterloo C&O Joint work with Isaac Fung TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAA

What are sparsifiers? • Weighted subgraphs that approximately preserve some properties [BSS’09] • Approximating all cuts • Sparsifiers: number of edges = O(n log n /²2) ,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 /²2), “entire spectrum” approximated within 1+². [SS’08] • O~(m) time algorithm to construct them n = # vertices Poly(n) m = # edges [BSS’09] Laplacian matrix of G Laplacian matrix of Sparsifier Poly(n)

n = # verticesm = # edges Why are sparsifiers useful? • Approximating all cuts • Sparsifiers: fast algorithms for cut/flow problem v = flow value

Our Motivation • BSS algorithm is very mysterious, and“too good to be true” • Are there other methods to get sparsifiers with only O(n/²2) edges? • Wild Speculation: Union of O(1/²2) random spanning trees gives a sparsifier(if weighted appropriately) • True for complete graph [GRV ‘08] • We prove: Speculation is false, butUnion of O(log2 n/²2) random spanning trees gives a sparsifier

n = # verticesm = # edges Formal problem statement • Design an algorithm such that • Input: An undirected graph G=(V,E) • Output: A weighted subgraph H=(V,F,w),where FµE and w : F !R • Goals: • | |±G(U)| - w(±H(U)) | ·² |±G(U)| 8U µ V • |F| = O(n log n / ²2) • Running time = O~( m / ²2 ) # edges between U and V\U in G weight of edges between U and V\U in H • | |±(U)| - w(±(U)) | ·² |±(U)| 8U µ V

Sparsifying Complete Graph • Sampling: Construct H by sampling every edge of Gwith 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)| 8U µ V • So H is a sparsifier of G

Generalize to arbitrary G? 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), parameters pe2 [0,1] • Output: A weighted subgraph H=(V,F,w),where FµE and w : F !R • For i=1 to ½ • For each edge e2E • With probability pe, Add e to F Increase we by 1/(½pe) • Main Question: Can we choose ½ and pe’sto achieve sparsification goals?

Non-uniform sampling algorithm [BK’96] • Input: Graph G=(V,E), parameters pe2 [0,1] • Output: A weighted subgraph H=(V,F,w),where FµE and w : F !R • For i=1 to ½ • For each edge e2E • With probability pe, Add e to F Increase we by 1/(½pe) • Claim: H perfectly approximates G in expectation! • For any e2E, E[ we ] = 1 • ) For every UµV, E[ w(±H(U)) ] = |±G(U)| • Goal: Show every w(±H(U)) is tightly concentrated

Assume ² is constant Prior Work Similar to edge connectivity • Benczur-Karger ‘96 • Set ½ = O(log n), pe = 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 • epe·n ) |F| = O(n log n) (# edges in sparsifier) • Running time is O(m log3 n) • Spielman-Srivastava ‘08 • Set ½ = O(log n), pe = “effective resistance” 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 • epe=n-1 ) |F| = O(n log n) (# edges in sparsifier) • Running time is O(m log50 n) • Uses “Matrix Chernoff Bound” O(m log3 n)[Koutis-Miller-Peng ’10]

Assume ² is constant Our Work • Fung-Harvey ’10 (independentlyHariharan-Panigrahi ‘10) • Set ½ = O(log2 n), pe = 1/edge-connectivity of edge e • All cuts are preserved • epe·n ) |F| = O(n log2 n) • Running time is O(m log2 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 (min size of a cut that contains e) Why? Pr[ e 2 T ] = effective resistance of eand edges are negatively correlated

Assume ² is constant Our Work • Fung-Harvey ’10 (independentlyHariharan-Panigrahi ‘10) • Set ½ = O(log2 n), pe = 1/edge-connectivity of edge e • All cuts are preserved • epe·n ) |F| = O(n log2 n) • Running time is O(m log2 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 log2 n) + O~(n) (min size of a cut that contains e) O(n log n)

Assume ² is constant Our Work • Fung-Harvey ’10 (independentlyHariharan-Panigrahi ‘10) • Set ½ = O(log2 n), pe = 1/edge-connectivity of edge e • All cuts are preserved • epe·n ) |F| = O(n log2 n) • Running time is O(m log2 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 /²2) edges, with running timeO(m) in unwtd graphs and O(m)+O~(n/²2) in wtd graphs (min size of a cut that contains e)

Notation: kuv = min size of a cut separating u and v • Main ideas: • Partition edges into connectivity classesE = E1[ E2 [ ... Elog n where Ei = { e : 2i-1·ke<2i }

Notation: kuv = min size of a cut separating u and v • Main ideas: • 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 is about right • This yields a sparsifier U

Notation: • C = ±(U) is a cut • Ci= ±(U) ÅEi is a cut-induced set • Need to prove: Prove weight of sampled edges that each cuttakes from each connectivity class is about right C2 C3 C1 C4

Notation:Ci= ±(U) ÅEi is a cut-induced set Prove 8 cut-induced set Ci • Key Ingredients • Chernoff bound: Provesmall • Bound on # small cuts: Prove #{ cut-induced sets Ciinduced by a small cut |C| }is small. • Union bound: sum of failure probabilities is small,so probably no failures. C2 C3 C1 C4

Counting Small Cut-Induced Sets • Theorem: 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: 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 }| < n2®.

Comparison (Slightly unfair) • Theorem: 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 |{ ±(U) ÅB : |±(U)|·c }| < n2c/K8c¸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 }| < n2c/K 8c¸1. • How many cuts of size 1? Theorem says < n2, taking K=c=1. Corollary, says < 1, because K=0.

Conclusions • Graph sparsifiers important for fast algorithms and some combinatorial theorems • Sampling by edge-connectivities gives a sparsifierwith O(n log2 n) edges in O(m log2 n) time • Improvements: O(n log n) edges in O(m) + O~(n) time[Panigrahi ‘10] • Sampling by effective resistances also works ) sampling O(log2 n) random spanning trees gives a sparsifier Questions • Improve log2 n to log n? • Sampling o(log n) random trees gives a sparsifier with o(log n) approximation?

Graph Sparsifiers by Edge-Connectivity and Random Spanning Trees