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

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

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

4. 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

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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

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

14. 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

15. 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

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

17. 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? …

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

19. General setup Matrix Sample N rows Pr[sum of rows= …
Matrix
Sample N rows
Pr[sum of rows=
expected sum of rows]
When can we guarantee it is positive?

20. General setup [Alon-Vu’97] example of regular …
[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

21. Main Theorem Pr[sum of N rows= expected sum of rows]>0 …
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

22. Main Theorem Some notation A – set of columns …
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

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

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

25. Main Theorem Main condition: column span V = space spanned by columns …
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

26. Main Theorem V = space spanned by columns Example: t-(n,k,) designs …
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

27. Main Theorem V = space spanned by columns Example: t-(n,k,) designs …
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

28. Main Theorem V = space spanned by columns Example: t-(n,k,) designs …
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

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

30. Main Theorem B x A matrix, V=span(columns) Assume … 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)

31. Conditions on V as a code (over the rationals)
Main Theorem … 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)

32. 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

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

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

35. Proof of main theorem X = sum of selected rows Pr[X=E[X]] = ? … A B
X = sum of selected rows
Pr[X=E[X]] = ?
Main tool: Fourier analysis
Fourier coefficients of X:

36. Proof of main theorem X = sum of selected rows … A B
X = sum of selected rows
Fourier coefficients of X:
Maximal Fourier coefs form a lattice:

37. Proof of main theorem
Fourier space
Maximal Fourier coefs

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

39. 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

40. 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

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

42. 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

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

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

45. 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