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The space complexity of approximating the frequency moments
It turns out that the numbers F0;F1 and F2 can be approximated in logarithmic space, whereas the approximation of Fk for k 6 requires n (1) space.
Quantum speed-up of Markov chain based algorithms
  • M. Szegedy
  • Computer Science
    45th Annual IEEE Symposium on Foundations of…
  • 17 October 2004
It is shown that under certain conditions, the quantum version of the Markov chain gives rise to a quadratic speed-up, and that the quantum escape time, just like its classical version, depends on the spectral properties of the transition matrix with the marked rows and columns deleted.
Proof verification and hardness of approximation problems
The authors improve on their result by showing that NP=PCP(logn, 1), which has the following consequences: (1) MAXSNP-hard problems do not have polynomial time approximation schemes unless P=NP; and (2) for some epsilon >0 the size of the maximal clique in a graph cannot be approximated within a factor of n/sup ePSilon / unless P =NP.
Proof verification and the hardness of approximation problems
It is proved that no MAX SNP-hard problem has a polynomial time approximation scheme, unless NP = P, and there exists a positive ε such that approximating the maximum clique size in an N-vertex graph to within a factor of Nε is NP-hard.
On the degree of boolean functions as real polynomials
A tight lower bound is found of Ω(logn) on the degree needed to represent any Boolean function that depends onn variables on the level of real polynomial.
Checking computations in polylogarithmic time
WJe show that every nondeterministic computational task S(Z, y), defined as a polynomial time relation between the instance x, representing the input and output combined, and the witness y can be modified to a task S such that each instance/witness pair becomes checkable in poly!ogariihmic Monte Carlo time.
Tracking join and self-join sizes in limited storage
This paper presents a join signature scheme based on tug-ofwar signatures that probvides guarantees on join size estimation as a function of t:he self-join sizes of the joining relations; this scheme can significantly improve upon the sampling scheme.
Interactive proofs and the hardness of approximating cliques
The connection between cliques and efficient multi-prover interaction proofs, is shown to yield hardness results on the complexity of approximating the size of the largest clique in a graph.
Approximating clique is almost NP-complete
The computational complexity of approximating omega (G), the size of the largest clique in a graph G, within a given factor is considered. It is shown that if certain approximation procedures exist,