Probabilistic existence of regular combinatorial objects Shachar Lovett (UCSD) Joint with Greg Kuperberg (UC Davis), Ron Peled (Tel-Aviv university)

Overview Regular combinatorial objects Probabilistic model Main Theorem: random walks on lattices Proof: Fourier analysis and codes Summary and open problems

Overview Regular combinatorial objects Probabilistic model Main Theorem: random walks on lattices Proof: Fourier analysis and codes Summary and open problems

Regular objects “highly symmetric” objects Regular graphs Regular hyper-graphs (aka designs) K-wise permutations Orthogonal arrays q-analogs … Constructions known in some special cases This work: First existence proofs for (nearly) all underlying parameters by a probabilistic argument

Regular graphs (n,d) regular graph – n vertices, all of degree d Easy to construct

Regular hyper-graphs Also known as designs t-(n,k,) design: k-uniform hyper-graph on n vertices; any t vertices belong to exactly  edges 1-(n,2,d) design: regular graph

Regular hyper-graphs t-(n,k,) design: k-uniform hyper-graph on n vertices; any t vertices belong to exactly  edges Constructions: Small values based on group symmetries [Teirlinck’87] first asymptotic construction of t-(n,t+1,) designs for infinitely many n,  Few other asymptotic constructions

Regular hyper-graphs t-(n,k,) design: k-uniform hyper-graph on n vertices; any t vertices belong to exactly  edges Existence unknown for most parameters: Steiner systems: t-(n,k,1) designs, open for t>5 Hadamard matrices: 2-(4m-1,2m-1,m-1) designs In general, constructions (and existence) unknown for almost all parameters

K-wise permutations 125346 251643 361254 Family of permutations acting uniformly on k elements A set FSn is k-wise if for any k distinct elements i1,…,ik and j1,…,jk

K-wise permutations 125346 251643 361254 Family of permutations acting uniformly on k elements Constructions: Subgroups of Sn: k=1,2,3 (e.g. for k=2 and n prime, subgroup of affine maps {x->ax+b}) Fail for k>3 (only Sn or An are 4-wise for n>24) [Finucane-Peled-Yaari’12] Combinatorial constructions for small k of exponential size

Other examples Orthogonal arrays: subsets of [c]n where any k coordinates get all values equally often (aka k-wise independent functions [n]->[c]) q-analogs: Family of k-dimensional subspaces of Fqn which cover uniformly all the t-dimensional subspaces (eg designs for the Grassmanian) Spherical designs: family of points on Sn which allow to integrate low degree polynomials by summing over the points

Our approach Probabilistic construction General technique to prove existence of regular objects Prove existence of designs, k-wise permutations, orthogonal arrays, for (nearly) all underlying parameters; of optimal size up to polynomial overhead

Overview Regular combinatorial objects Probabilistic model Main Theorem: random walks on lattices Proof: Fourier analysis and codes Summary and open problems

Probabilistic model Running example: t-(n,k,) designs k-uniform hyper-graph on n vertices; any t vertices belong to exactly  edges Random construction: Sample N=N(n,k,t,) edges uniformly Analyze probability that any t vertices covered by exactly  edges Very unlikely event

Probabilistic model Random constructions unlikely to work But is probability zero or tiny but positive? How can we analyze “rare events” ? Standard tools fail, e.g. Limited dependence (e.g. Lovasz Local lemma) doesn’t hold Spencer’s method not relevant

Probabilistic model Another viewpoint: sum of matrix rows t-subsets of [n] Incidence matrix 0100101001 1001011000 0000101110 … 0010110100 Sample few rows Analyze probability that sum is (,…,) Pr[sum=expected sum] Edges: k-subsets of [n]

Probabilistic model Yet another viewpoint: short random walk on a lattice Lattice spanned by rows Steps: rows Probability that a short random walk will end in a specific point Can we guarantee fast convergence? 0100101001 1001011000 0000101110 … 0010110100

Overview Regular combinatorial objects Probabilistic model Main Theorem: random walks on lattices Proof: Fourier analysis and codes Summary and open problems

General setup Matrix Sample N rows Pr[sum of rows= 0100101001 1001011000 0000101110 … 0010110100 Matrix Sample N rows Pr[sum of rows= expected sum of rows] When can we guarantee it is positive?

General setup [Alon-Vu’97] example of regular 0100101001 1001011000 0000101110 … 0010110100 [Alon-Vu’97] example of regular hyper-graphs on n vertices, ~nn/2 edges, with no regular sub-hypergraphs Pr[sum of rows= expected sum of rows]=0 Conclusion: need to assume some structure

Main Theorem Pr[sum of N rows= expected sum of rows]>0 0100101001 1001011000 0000101110 … 0010110100 Main theorem: set of conditions that guarantee that N is polynomial in underlying parameters In our applications we get optimal N (up to polynomial factors) Can approximate probability up to 1+o(1) Pr[sum of N rows= expected sum of rows]>0

Main Theorem Some notation A – set of columns 0100101001 1001011000 0000101110 … 0010110100 Some notation A – set of columns B – set of rows (|B| >> |A|) V – linear space spanned by columns row(b) – row in index bB We want SB of size |S|=N such that B

Main Theorem Pre-condition: divisibility We want |S|=N for which 0100101001 1001011000 0000101110 … 0010110100 Pre-condition: divisibility We want |S|=N for which Let c1 be minimal integer such that N must be a multiple of c1 B

Main Theorem Pre-condition: divisibility Example: t-(n,k,) designs 0100101001 1001011000 0000101110 … 0010110100 Pre-condition: divisibility Example: t-(n,k,) designs [Wilson’73, Graver-Jurkat’73] analyze divisibility of incidence matrices N multiple of B

Main Theorem Main condition: column span V = space spanned by columns 0100101001 1001011000 0000101110 … 0010110100 Main condition: column span V = space spanned by columns Need: (a) V has transitive symmetry group (b) V spanned by short integer vectors in l (c) V spanned by short integer vectors in l1 (in coding terms, V is an LDPC) (d) V contains the constant vectors B

Main Theorem V = space spanned by columns Example: t-(n,k,) designs 0100101001 1001011000 0000101110 … 0010110100 V = space spanned by columns Example: t-(n,k,) designs (a) V has transitive symmetry group Sn acts as permutations on k-subsets (rows) and t-subsets (columns) Acts transitively on rows (e.g. B) B

Main Theorem V = space spanned by columns Example: t-(n,k,) designs 0100101001 1001011000 0000101110 … 0010110100 V = space spanned by columns Example: t-(n,k,) designs (b) V spanned by short integer vectors in l Immediate since matrix has small elements, so columns are such a basis for V B

Main Theorem V = space spanned by columns Example: t-(n,k,) designs 0100101001 1001011000 0000101110 … 0010110100 V = space spanned by columns Example: t-(n,k,) designs (c) V spanned by short integer vectors in l1 Usually the hardest condition to verify; for designs, requires some combinatorial calculations B

Main Theorem V = space spanned by columns Example: t-(n,k,) designs 0100101001 1001011000 0000101110 … 0010110100 V = space spanned by columns Example: t-(n,k,) designs (d) V contains the constant vector Sum of columns is constant B

Main Theorem B x A matrix, V=span(columns) Assume 0100101001 1001011000 0000101110 … 0010110100 B B x A matrix, V=span(columns) Assume c1 divisibility constant V spanned by integer vectors with l bound c2 V spanned by integer vectors with l1 bound c3 V has transitive symmetry group V contains the constant vectors Then for N=poly(|A|,c1,c2,c3), Pr[sum of N rows= expected sum]>0 In fact, we approximate the probability up to 1+o(1)

Conditions on V as a code (over the rationals) Main Theorem 0100101001 1001011000 0000101110 … 0010110100 B B x A matrix, V=span(columns) Assume c1 divisibility constant V spanned by integer vectors with l bound c2 V spanned by integer vectors with l1 bound c3 V has transitive symmetry group V contains the constant vectors Then for N=poly(|A|,c1,c2,c3), Pr[sum of N rows= expected sum]>0 In fact, we approximate the probability up to 1+o(1) Conditions on V as a code (over the rationals)

Applications Optimal size up to polynomial overhead Can count the number of objects (up to 1+o(1)) Existence of t-(n,k,) designs with N=(n/t)O(t) Verification of conditions relatively simple Existence of k-wise permutations with N=nO(k) permutations Verification of conditions was harder; required nontrivial representation theory of Sn Orthogonal arrays

Overview Regular combinatorial objects Probabilistic model Main Theorem: random walks on lattices Proof: Fourier analysis and codes Summary and open problems

Proof of main theorem N - Target #rows 0100101001 1001011000 0000101110 … 0010110100 A B N - Target #rows Choose each row with prob p=N/|B| X = sum of selected rows (random var) Pr[X=E[X]] = ?

Proof of main theorem X = sum of selected rows Pr[X=E[X]] = ? 0100101001 1001011000 0000101110 … 0010110100 A B X = sum of selected rows Pr[X=E[X]] = ? Main tool: Fourier analysis Fourier coefficients of X:

Proof of main theorem X = sum of selected rows 0100101001 1001011000 0000101110 … 0010110100 A B X = sum of selected rows Fourier coefficients of X: Maximal Fourier coefs form a lattice:

Proof of main theorem Fourier space Maximal Fourier coefs

Proof of main theorem Fourier space Maximal Fourier coefs Step I: approximate F.C. near maximas Gaussian approximation Relatively straight-forward

Proof of main theorem Fourier space Maximal Fourier coefs Step II: bound F.C. far from maximas Most Fourier space Harder step, requires all the conditions on V Develop new connections between coding properties of V and the Fourier coefs

Proof of main theorem End result: Gaussian approximation, restricted to the lattice in which X lives X = sum of N rows Y = Gaussian with same covariance as X Pr[X=E[X]]  density of Y at E[X] times some lattice related factors

Overview Regular combinatorial objects Probabilistic model Main Theorem: random walks on lattices Proof: Fourier analysis and codes Summary and open problems

Summary New probabilistic technique Theorem: can prove existence of regular structures by verification of a few conditions, which are verified explicitly This is in contrast to the existence result which is non-explicit

Summary Proof technique: Fourier analysis Make new connections between coding theory and Fourier analysis in order to bound Fourier coefficients

Open problems Applications Work in progress (with Kuperberg and Peled): spherical designs Work in progress (with Vardy): q-analogs

Open problems Algorithms Current proof is purely existential Don’t know how to find objects efficiently, even using randomness Other probabilistic techniques for rare events were made algorithmic, so there is hope Lovasz local lemma: Moser, Moser-Tardos Spencer’s theorem: Bansal, L-Meka