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Complexity of Graph Properties via Randomized Decision Trees

This paper examines the computational complexity of graph properties using randomized decision trees. It explores various graph properties such as connectivity, Hamiltonicity, and k-cliques, detailing deterministic and randomized complexities for decision trees. Notable results include the threshold probabilities and the connection between subcube partitions and decision trees. The authors discuss the implications of these findings on graph properties, establishing bounds for deterministic and randomized complexities, and addressing open problems in the field.

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Complexity of Graph Properties via Randomized Decision Trees

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  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(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 0 1 x1 x3 0 0 1 1 x1 0 1 1 0 1 1 0

  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 1/n n2/log n has 4-clique 1/n2/3 n5/3 (2log n)-clique 1/2 n

  8. 1 1 0 0 1 0 1 0 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)

  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)

  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=1 1(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

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