1. Computing Graph Properties by Randomized Subcube Partitions
Ehud Friedgut
Hebrew university
Jeff Kahn
Rutgers University
Avi Wigderson
IAS & Hebrew U.

2. Decision Trees f:{0,1}N{0,1} T T(x) path taken by x
T computes f iff for
every x, v(T(x))=f(x)
cost(T) = maxx |T(x)|
DET(f) = minT cost(T) T x2 1 x1 x3 1 1 x1 1 1 1 1

3. Graph Properties N = n(n-1)/2 x encodes an n-vertex graph G
f is a graph property if f(x) depends
only on the isomorphism type of G
Monotone, nontrivial graph properties
connected, planar, Hamiltonian, empty,
contains a k-clique, k-colorable, …

4. Deterministic Complexity of Graph Properties
Thm [RV] For every graph property f and every n, DET(f) = Ω(n2)
Thm [KSS] For every graph property f and even prime n, f is evasive:
DET(f) = N = n(n-1)/2
Thm [CKS] Some natural families of graph properties are evasive for every n.

5. Probabilistic Decision Trees
T a random variable over trees T for f
Cost(T ) = maxx ET [|T(x)|]
RAND(f) = minT cost(T )
Thm [SW] There exists f:{0,1}N{0,1}
with DET(f) = N, RAND(f) = N.753
Moreover, f is transitive.

6. Probabilistic Complexity of Graph Properties
Conj [K] RAND(f) = Ω(n2)
Thm [Y,K, H, CK] RAND(f) = Ω(n4/3log1/3n)
New RAND(f) = Ω ( min {n2/log n, n/p(f) } )
where PrG(n,p(f)) [f(G)=1] = 1/2

7. Threshold Probabilities
Property p(f) RAND(f)>
Connectivity,
Hamiltonicity (log n)/n n2/log n
has a triangle /n n2/log n
has 4-clique /n2/ n5/3
(2log n)-clique / n
But we know n^2 a different way

8. Subcube Partitions C = C1 C2  …  Ct partition {0,1}N Ci {0,1,*}N
|Ci |= codim = no. of 0/1’s
C(x) – the subcube Ci containing x
cost(C) = maxx |C(x)|
C computes f: f is constant on subcubes
DETS(f), RANDS(f) as before.
Fact DETS(f)DET(f), RANDS(f)  RAND(f) 1

9. Plan of Proof Thm For every graph property f,
RANDS(f) > Ω(min {n2/log n, n/p(f) })
Proof p=p(f) satisfies (log n)/n < p < ½
Pick a graph x at random from G(n,p)
Prove that for any C computing f,
RANDS(f) > E[|C(x)|] > Ω(n/p)
Say we use Yao’s minimax

10. Plan of Proof (continued)
1(x) – number of 1’s in C(x)
0(x) – number of 0’s in C(x)
Fact |C(x)|=1(x)+0(x)
Lemma E[0(x)] > Ω(n/p)
Proof
Case 1: E[1(x)] < n/8 packing argument (using f)
Case 2: E[1(x)] > n/8 independent of f

11. Graph Packing Claim E[1(x)] < n/64  E[0(x)] > Ω(n/p)
Thm [C,SS] G,H two graphs on V
(G)(H) < |V|/2  G & H pack
E[1(x)] < n/8  G’ f(G’)=1, (G) < 2np, G’ has < n/4 edges
E[0(x)] < n/64p  H’ f(H’)=0, H’ has < n/32p edges.
Packing G’ and H’
Take V[n], |V|=n/2, G=G’ on V
Map all vertices of H’ of deg > 1/16p to Vc
On V, H remainder of H’. (G)(H) < (2np)(1/8p) = n/4 = |V|/2

12. Product Distributions and Subcube Partitions
Claim E[1(x)] > n/8  E[0(x)] > Ω(n/p)
Thm C any subcube partition of {0,1}N
x{0,1}N drawn at random with
xi=1 indep with prob p. Then
E[0(x)]/E[1(x)] = (1-p)/p > 1/2p.
Proof 0j(x)=1 if C(x) forces xj=0 0(x)=j 0j(x)
1j(x)=1 if C(x) forces xj= (x)=j 1j(x)
j[n], E[0j(x)]/E[1j(x)] = (1-p)/p

13. Open Problems Find f which exhibit large gaps between
DETS(f) &DET(f), RANDS(f) & RAND(f)
P(f)=1/2, G random in G(n,1/2).
Conj E[|witness(G)|] = Ω(n2/ log n)
(G)= prob G appears in G(n,1/2)
Fact (G)>1/2, (H)>1/2  G & H pack
Conj >0, (G)>, (H)>  G & H pack