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Cuts, Trees, and Electrical Flows Aleksander Mądry

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Graphs are everywhere! (In case you just got here by mistake…) They got HUGE

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(Could allow directed edges, but will focus on undirected today) Algorithmic Graph Theory: Developing algorithmic ideas and tools to allow us to analyze and understand graphs

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Example Tasks of Interest Graph Partitioning (clustering, community detection, div & conq) Connectivity Analysis (congestion estimation, analyzing resiliance to link failures) Network Design (supporting most efficent commun. infrastructure, routing schemes)

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(Could allow directed edges, but will focus on undirected today) Algorithmic Graph Theory: Developing algorithmic ideas and tools to allow us to analyze and understand graphs Big Data = Big Problem We need our algorithms to be really fast (theory paradise: asymptotic complexity does matter here, O(n 2 ) or even O(n 3/2 ) does not cut it) Also, parallelization/distributed aspects play role Classic approaches that we used so far are often too slow and hard to parallelize/decentralize

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Fundamental question: What graph problems can we solve in nearly-linear time?

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Undirected graph G with integer capacities u() The setup We want to solve minimization cut-based problems on G

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Cut-based min problem C u(C) = capacity of the cut C f(C) = arbitrary positive function independent of edges/capacities f(C) = 1 minimum cut problem f(C) = 1 if C separates s and t; f(C)= otherwise minimum s-t cut problem f(C) = 1/min {|C|, |V\C|} sparsest cut problem Examples:

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generalized sparsest cut problem minimum conductance cut problem balanced separator problem minimum bisection problem minimum multi-cut problem minimum multiway-cut problem … Other important examples: Can we solve these problems in poly time? Yes, in case of minimum (s-t) cut, but the remaining ones are NP-hard Good news: Can get a O(log n) (or better) approx.!

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What about nearly-linear time computations? Bad news: Known alg. for min s-t cut need Ω(n 3/2 ) time Same is true for (almost) all the best approx. algorithms for the other problems How about trading approx. for efficiency? (We already did this to cope with NP-hardness) Good news: Possible for minimum cut problem! [Karger 00] Lots of progress in the context of sparsest cut and minimum conductance cut [Arora Hazan Kale 04 Khandekar Rao Vazirani 06 Arora Kale 07 Orecchia Schulman Vazirani Vishnoi 08 Sherman 09 Spielman Teng 04 Andersen Chung Lang 06 Andersen Peres 09 Orecchia Vishnoi 11 Orecchia Sachdeva Vishnoi 12]

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Still-unanswered question: Can we design nearly-linear time approx. algorithms for all these problems? [M. 10]: Yes, if we can compute such a fast approx. on trees More precisely: Fix any cut-based minimization problem Good and fast β-approx. alg. working only on trees A O(β polylog)-approx. for general graphs that runs in Õ(n 1+δ ) time (The poly-log grows mildly with 1/δ) Moral: When shooting for a fast polylog-approx. algorithm for a cut-based minimization problem, just focus on trees

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G C C C λ1λ1 λiλi (T i,λ i ) (T 1,λ 1 ) How to prove such a theorem? such that for any cut C: (cut lower-bounding) u i (C) u(C) for all i (cut upper-bounding) E λ [u(C)]:= i λ i u i (C) O(log n) u(C) [Räcke 08] (simplified): For any graph G=(V,E,u), we can find a convex combination {(λ i,T i )} i of trees*

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How to use this decomposition? Idea for lifting: 1) Find {(λ i,T i )} i as described 2) Sample a tree T being T i with prob. λ i 3) Output a β-optimal solution C for instance P on T β u T (C*)f P (C*) O(β log n) u(C*) f P (C*)=O(β log n) OPT Why should it work? With prob. 1/2: u T (C*) O(log n) u(C*) Let C* be the optimal solution But u(C) f P (C) u T (C) f P (C) Note: Choice of T is oblivious to the problem we want to solve

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Lifting works great! How about running time? Räckes algorithm runs in Õ(n 3 ) time Prohibitive from our point of view! What to do now? Idea: Decompose G into objects that are more complicated than trees, but still simpler than general graphs H is a j-tree if it is a union of: a forest F (envelope) an arbitrary graph R (core) and: 1) |V(R)| j 2) for each connected component F of F, |V(F) V(R)|=1 Note: 1-tree is just a tree H R F

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Decomposing graphs into j-trees The real benefit comes from the ability to vary j! Theorem (simplified): For any graph G=(V,E,u) and j1, we can find in Õ(m+n 2 /j) time a convex comb. {(λ i,T i )} i of j-trees s.t. for any cut C: (cut lower-bounding) u i (C) u(C) for all i (cut upper-bounding) E λ [u(C)]:= i λ i u i (C) Õ(log n) u(C) j=1 yields Räckes result with faster running time but has slightly worse quality

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G Speeding up min s-t cut computation: Õ(m+n 2 /j) time + sampling C* T R With prob. 1/2, cut C* is Õ(log n)-preserved If we run the min s-t cut alg. on T instead of G then appropriate choice of j gives a speed up! (i.e. if j is such that we minimize the total running time: Õ(m+n 2 /j) + Õ(m+j (1+c) ) << Õ(m+n 1+c )) s t s t

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G Õ(m+n 2 /j) time + sampling C* T R s t s t We get a running time arbitrarily close to nearly-linear There is even better way of leveraging this flexibility! …we do it in a series of small recursive steps …but at a price of approx. ratio growing accordingly Instead of reducing G to T in one big step…

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Success? We get a good generic baseline solution for a large family of problems But… The approx. ratio could be better Algorithms seem not really practical yet (Can we use some of the insight to speed up the existing heuristics?) Still… The approx ratio should be improvable Was recently used to get (1+ε)-approx. to undirected max flow in close-to-linear time [Kelner Lee Orecchia Sidford 13] [Sherman 13]

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Spectral Graph Theory Beyond λ 2

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= L Spectral graph theory: Connecting combinatorial structure of G to linear-algebraic properties of L The Laplacian matrix of a graph G AD- D = diagonal vertex degree matrix A = adjacency matrix Key quantity: λ 2 = second-smallest eigenvalue of L

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Classic success story: λ 2 connection Lots of connections to spectral properties of the graph (e.g., understanding of random walks) Spectral (and local) partitioning: Works great on well-connected graphs (i.e., graphs with large λ 2 ) But has well-known limitations Can we take spectral graph theory beyond λ 2 ?

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Promising example: O(k 2 λ 2 (λ k ) -1/2 )-approx. to uniform sparsest cut [Kwok Lau Lee Oveis-Gharan Trevisan 13] (If λ k fairly large for some small k a very good approx.) Upshot: λ k fairly large G = union of k-1 expander-like graphs Still: Can we access the whole spectrum of L in more principled manner?

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Given a graph G with resistances {r e } e source s and sink t s t resistances given by {r e } e Recipe for elec. flow: 1) Treat edges as resistors 2) Connect a battery to s and t Object of interest: Electrical flows

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Given a graph G with resistances {r e } e source s and sink t s t Recipe for elec. flow: 1) Treat edges as resistors 2) Connect a battery to s and t induced current (flow conservation constraints hold) Object of interest: Electrical flows

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Computing an electrical flow = solving a Laplacian linear system = L x b What does it have to do with Laplacians? Furthermore: [Spielman Teng 04, Koutis Miller Peng 10 11] We can (essentially) solve such systems in nearly-linear time Result: Electrical flow is a nearly-linear time primitive

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Fast maximum/multi-commodity flow approximation [Christano Kelner M. Spielman Teng 11] [Kelner Miller Peng 12] How can we employ this primitive? Faster random walk simulations [Kelner M. 09] Where else will electrical flow be a useful primitive?

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Conclusion Power of tree-like decompositions: Approx. vs. running time trade-off for undirected cut-based minimization problems Outstanding challenge: Directed graphs Spectral graph theory beyond λ 2 : Electrical flows primitive

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Thank you Questions?

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