Happy 60 th B’day Nati

Nati’s long view Avi Wigderson Institute for Advanced Study 2013 was a good year…

Analysis in Boolean complexity Local vs. Global phenomena Locality-sensitive hashing & BIG DATA Approximation theory in combinatorics Physics for graph algorithms Geometry in graphs and approx algorithms Geometry in Comm. Comp. & learning Cryptography without hardness Instance-based complexity Expander graphs: lifts, high dim, book… Deterministic approximation of Permanent Seeds, saplings, trees, forests

Geometry, Graphs, Approx algs [Linial-London-Rabinovich ’95] Many combinatorial and algorithmic problems concern either directly or implicitly the distance, or metric on the vertices of a possibly weighted graph. It is therefore natural to look for connections between such questions and classical geometric structures. The approach taken here is to model graph metrics by mapping the vertices to a real normed spaces in such a way that the distance between any two vertices is close to the distance between their geometric images. Embeddings are sought so as to minimize (i) the dimension of the host space, and (ii) the distortion, i.e., the extent to which the combinatorial and geometric metrics disagree. Specifically, we ask: 1.How small can the dimension and the distortion be for given graphs? Is it computationally feasible to find good embeddings? 2.Which graph-algorithmic problems are easier for graphs with favorable (low- dimensional, small distortion) embeddings? What are the combinatorial implications of such embeddings? 3.The above discussion extends to embeddings of general finite metric spaces. What are the algorithmic and combinatorial implications in this more general context?

Metric geometry, Banach space theory (X,d), (Z,d’) metric spaces. φ :X  Z is a distortion C embedding if for all x,y in X d(x,y) ≤ d’( φ (x), φ (y)) ≤ Cd(x,y) C p (X) – best possible distortion into L p [Bourgain] C 2 (X,d) ≤ log n for every |X|=n [Enflo] C 2 ({0,1} k,h) ≈ √k ( k = log n ) (X,d) is of Negative Type if (X,√d) is in L 2 Metric embeddings

Multi-commodity flow / sparsest cut problem G graph on [n], Capacity, Demand: [n]x[n]  R Find largest λ s.t. min-cut ≤ λ max-flow [Leighton-Rao] λ ≤ log n, efficiently! [LLR] -This is a problem in metric geometry: λ = best distortion of embedding a metric in L 1 -Alg version of Bourgain’s embedding thm  [LR] -Tightness of Bourgain via expander metrics -Semi-definite formulation of embedding to L 2 -Alg & structural consequences  800+ papers [ARV,ALN] c 2 (Neg) ≤ √log n, efficiently! [LG,KV] c 1 (Neg) ≥ (log log n) α via hardness! Geometry and approx algs [Linial-London-Rabinovich ’95]

Why does Simplex works well in practice? How can our body optimally fold proteins? What are “naturally arising” instances? [BBG,BL] Practically arising instances are “robust”. Can robust instances be solved efficiently? [BL] Instance is γ –robust if changing “weights” (dist,similarity) by ≤ γ factor fixes optimal solution Max Cut. Polytime alg: [BL] γ ≤n, [BDLS] γ ≤√n, [MMV] γ ≈ c 1 (Neg) (“duality” with sparsest cut) Stable instances of hard problems [Bilu-Linial ’10]

Expanders are GREAT [Hoory-Linial-Wigderson] d-regular graphs. λ -expander: all nontrivial e-vals ≤ λ [Pinsker] Random d-reg graphs are.9d-expanders [Margulis] First explicit construction. Algebraic! [M,PLS] Explicit d-reg Ramanujan (2√(d-1))-exp. [F] λ ≤ 2√(d-1)+ ε for almost all d-reg graphs. [AB] λ ≥ 2√(d-1) (Ramanujan is tight). [RVW] Combinatorial Zig-Zag expanders. λ ≤ d 2/3 [BL] Lets go for combinatorial construction of Ramanujan graphs! Expander from Lifts [Bilu-Linial ’06]

d-regular graphs. Ramanujan bound 2√(d-1)) is the spectral radius of the (infinite) d-regular tree, which is the universal cover of all such graphs! Climb the tree! Lifts: finite covers. Perform a sequence of “good” 2-lifts, start with a good exp (eg K d+1 ). [BL] Thm: λ ≤ √d log 3/2 d efficiently! [BL] Conj: λ ≤ 2√(d-1) (Ramanujan is tight). [MSS] conjecture is true. Ramanujan graphs of every degree! Interlacing polynomials,… [MSS] Kadison-Singer conj is true!! Expander from Lifts [Bilu-Linial ’06]

[Linial-Sassosn] Completely counterintuitive! Hashing – drastically reducing description size, keeping distinction. Highly mixing! New: keeping proximity! Minimizing Page faults Nearest neighbour search Compression, Sketching, Info retrieval Learning/Clustering/…… Locality-sensitive Hashing

H = { h: [U]  [R] } is good if mixing and neighborly (1) For all h,p,q, dist(h(p),h(q)) ≤ dist(p,q) Easy: h(p)=1 for all p. (2) Whp h bijective on small sets (<√R ) Easy: Pairwise indep. family H [LS] Achieve both! |H|=|U|, dynamic,… [IMRV] High dim. BIG data, Many alg apps. Locality-sensitive Hashing [Linial-Sassosn ‘98]

[LS] Factorization norm γ 2 for quantum cc, PAC,… Log rank conjecture [Lovasz-Saks]: M nxn sign matrix of rank r. [MS] log r < cc(M) < r [LS] cc(M) < (log r) c ?? [Nisan-W] disc(M) > 1/r 3/2 algebra [LMSS] disc(M) > 1/r 1/2 geometry [Lovett] cc(M) < √r Geometry in Comm Comp + Learning [L-Mendelson-Shraibman-Schectman ’07] [LS’09] cc=communication complexity disc=discrepancy

A couple of collaborations (and a couple of cool algorithms for basic problems on graphs )

[L-Lovasz-Wigderson] G graph, X a set of k vertices. φ : V(G)  R k-1 is non-degenerate, convex if - φ (X) is in general position -φ maps X to the k vertices of a simplex -φ maps every other vertex to the convex hull of its neighbors [LLW] G is k-connected iff every X has such φ Proof: Make edges springs of random strengths. Glue X to the simplex and let the system relax. Cor: Best known k-conn algorithm for large k. Convex embeddings of graphs

Per n (X) = Σ πε Sn Π i X i, π (i) Non-negative matrices [JSV] Prob. polytime (1+ ε )-approx for Per n [LSW] Det. polytime e n -approx for Per n [GS] Det. polytime e n -approx for Mixed-Det n [GS] Det. polytime 2 n -approx for Per n Open: Det. polytime (1+ ε )-approx for Per n Matrix scaling of a matrix A: Multiplying rows & cols by constants (numerical analysis, Imaging,…) Self-reduction for the Permanent. Det approx of Per [Linial-Samorodnitsky-Wigderson]

A: adjacency matrix of an nxn bipartite graph. Alg: Repeat n 3 times: Make all row-sums of A 1 Make all col-sums of A 1 [LSW] A has PM iff all row & col sums < 1+1/n 2 Other scaling application (NZ=NonZero) [Dvir-Saraf-W] A has Θ (√n) NZ’s per row, Θ (1) NZ’s per pair of rows, then A has rank Θ (n). Matrix Scaling (and a new perfect matching alg)

Happy 60 th b’day Nati!!!