Graph Sparsifiers Nick Harvey University of British Columbia Based on joint work with Isaac Fung, and independent work of Ramesh Hariharan & Debmalya Panigrahi TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A

Approximating Dense Objects by Sparse Objects Floor joists Wood JoistsEngineered Joists

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Approximating Dense Objects by Sparse Objects Graphs Dense GraphSparse Graph

Input: Undirected graph G=(V,E), weights u : E ! R + Output: A subgraph H=(V,F) of G with weights w : F ! R + such that |F| is small and u( ± G (U)) = (1 § ² ) w( ± H (U)) 8 U µ V weight of edges between U and V\U in Gweight of edges between U and V\U in H Graph Sparsifiers Weighted subgraphs that approximately preserve graph structure Cut Sparsifiers (Karger ‘94) GH

Input: Undirected graph G=(V,E), weights u : E ! R + Output: A subgraph H=(V,F) of G with weights w : F ! R + such that |F| is small and x T L G x = (1 § ² ) x T L H x 8 x 2 R V Spectral Sparsifiers Weighted subgraphs that approximately preserve graph structure (Spielman-Teng ‘04) Laplacian matrix of GLaplacian matrix of H GH

Motivation: Faster Algorithms Dense Input graph G Exact/Approx Output Algorithm A for some problem P Sparse graph H Approximately preserves solution of P Algorithm A runs faster on sparse input Approximate Output (Fast) Sparsification Algorithm Min s-t cut, Sparsest cut, Max cut, …

State of the art SparsifierAuthors# edgesRunning Time CutBenczur-Karger ’96O(n log n / ² 2 )O(m log 3 n) SpectralSpielman-Teng ’04O(n log c n / ² 2 )O(m log c n / ² 2 ) SpectralSpielman-Srivastava ’08O(n log n / ² 2 )O(m log c n / ² 2 ) SpectralBatson et al. ’09O(n / ² 2 )O(mn 3 ) CutThis paperO(n log n / ² 2 )O(m + n log 5 n / ² 2 ) * SpectralLevin-Koutis-Peng ’12O(n log n / ² 2 )O(m log 2 n / ² 2 ) n = # vertices m = # edges *: The best algorithm in our paper is due to Panigrahi. ~ c = large constant

Random Sampling Can’t sample edges with same probability! Idea: [Benczur-Karger ’96] Sample low-connectivity edges with high probability, and high-connectivity edges with low probability Keep this Eliminate most of these

Generic algorithm Input: Graph G=(V,E), weights u : E ! R + Output: A subgraph H=(V,F) with weights w : F ! R + Choose ½ (= #sampling iterations) Choose probabilities { p e : e 2 E } For i=1 to ½ For each edge e 2 E With probability p e Add e to F Increase w e by u e /( ½ p e ) E[|F|] · ½ ¢  e p e E[ w e ] = u e 8 e 2 E ) For every U µ V, E[ w( ± H (U)) ] = u( ± G (U)) [Benczur-Karger ‘96] Goal 1: E[|F|] = O(n log n / ² 2 ) How should we choose these parameters? Goal 2: w( ± H (U)) is highly concentrated

Benczur-Karger Algorithm Input: Graph G=(V,E), weights u : E ! R + Output: A subgraph H=(V,F) with weights w : F ! R + Choose ½ = O(log n / ² 2 ) Let p e = 1/“strength” of edge e For i=1 to ½ For each edge e 2 E With probability p e Add e to F Increase w e by u e /( ½ p e ) Fact 1: E[|F|] = O(n log n / ² 2 ) Fact 2: w( ± H (U)) is very highly concentrated ) For every U µ V, w( ± H (U)) = (1 § ² ) u( ± G (U)) “strength” is a slightly unusual quantity, but Fact 3: Can estimate all edge strengths in O(m log 3 n) time “strength” is a slightly unusual quantity Question: [BK ‘02] Can we use connectivity instead of strength?

Our Algorithm Input: Graph G=(V,E), weights u : E ! R + Output: A subgraph H=(V,F) with weights w : F ! R + Choose ½ = O(log 2 n / ² 2 ) Let p e = 1/“connectivity” of e For i=1 to ½ For each edge e 2 E With probability p e Add e to F Increase w e by u e /( ½ p e ) Fact 1: E[|F|] = O(n log 2 n / ² 2 ) Fact 2: w( ± H (U)) is very highly concentrated ) For every U µ V, w( ± H (U)) = (1 § ² ) u( ± G (U)) Extra trick: Can shrink |F| to O(n log n / ² 2 ) by using Benczur-Karger to sparsify our sparsifier!

Motivation for our algorithm Connectivities are simpler and more natural ) Faster to compute Fact: Can estimate all edge connectivities in O(m + n log n) time [Ibaraki-Nagamochi ’92] ) Useful in other scenarios Our sampling method has been used to compute sparsifiers in the streaming model [Ahn-Guha-McGregor ’12]

Overview of Analysis Most cuts hit a huge number of edges ) extremely concentrated ) whp, most cuts are close to their mean Most Cuts are Big & Easy!

Overview of Analysis High connectivity Low sampling probability Low connectivity High sampling probability Hits many red edges ) reasonably concentrated Hits only one red edge ) poorly concentrated The same cut also hits many green edges ) highly concentrated This masks the poor concentration above There are few small cuts [Karger ’94], so probably all are concentrated. Key Question: Are there few such cuts? Key Lemma: Yes!

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 – This yields a sparsifier U

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 Chernoff bounds can analyze each cut-induced set, but… Key Question: Are there few small cut-induced sets? C1C1 C2C2 C3C3 C4C4

Counting Small Cuts Lemma: [Karger ’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 ®. Example: Let G = n-cycle. Edge connectivity is K=2. Number of cuts of size c = £ ( n c ). ) |{ ± (U) : | ± (U)| · ® K }| · O (n 2 ® ).

Counting Small Cut-Induced Sets Our Lemma: 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 ®. Karger’s Lemma: the special case B=E and K=min cut.

When is Karger’s Lemma Weak? Lemma: [Karger ’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 c ¸ K, |{ ± (U) : | ± (U)| · c }| < n 2c/K. Example: Let G = n-cycle. Edge connectivity is K=2 |{ cuts of size c }| < n c ² K = ² < n 2c/ ²

Our Lemma Still Works Our Lemma: 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 ®. Example: Let G = n-cycle. Let B = cycle edges. We can take K=2. So |{ ± (U) Å B : | ± (U)| · ® K }| < n 2 ®. |{ cut-induced subsets of B induced by cuts of size · c }| · n c ²

Algorithm for Finding a Min Cut [Karger ’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.

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

Conclusions Questions Sampling according to connectivities gives a sparsifier We generalize Karger’s cut counting lemma Improve O(log 2 n) to O(log n) in sampling analysis Applications of our cut-counting lemma?