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

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

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, …

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.

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.

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

Threshold Probabilities Property p(f) RAND(f)> Connectivity, Hamiltonicity (log n)/n n2/log n has a triangle 1/n n2/log n has 4-clique 1/n2/3 n5/3 (2log n)-clique 1/2 n But we know n^2 a different way

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

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

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

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

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=1 1(x)=j 1j(x) j[n], E[0j(x)]/E[1j(x)] = (1-p)/p

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