Mario Szegedy

Proof verification and the hardness of approximation problems

We show that every language in NP has a probablistic verifier that checks membership proofs for it using logarithmic number of random bits and by examining a constant number of bits in the proof. If a string is in the language, then there exists a proof such that the verifier accepts with probability 1 (i.e., for every choice of its random string). For strings not in the language, the verifier rejects every provided “proof” with probability at least 1/2. Our result builds upon and improves a recent result of Arora and Safra [1998] whose verifiers examine a nonconstant number of bits in the proof (though this number is a very slowly growing function of the input length). As a consequence, we prove that no MAX SNP-hard problem has a polynomial time approximation scheme, unless NP = P. The class MAX SNP was defined by Papadimitriou and Yannakakis [1991] and hard problems for this class include vertex cover, maximum satisfiability, maximum cut, metric TSP, Steiner trees and shortest superstring. We also improve upon the clique hardness results of Feige et al. [1996] and Arora and Safra [1998] and show that 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.

The space complexity of approximating the frequency moments

The frequency moments of a sequence containing mi elements of type i, for 1 i n, are the numbers Fk = P n=1 m k . We consider the space complexity of randomized algorithms that approximate the numbers Fk, when the elements of the sequence are given one by one and cannot be stored. Surprisingly, 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. Applications to data bases are mentioned as well.

Proof verification and hardness of approximation problems

The class PCP(f(n),g(n)) consists of all languages L for which there exists a polynomial-time probabilistic oracle machine that used O(f(n)) random bits, queries O(g(n)) bits of its oracle and behaves as follows: If x in L then there exists an oracle y such that the machine accepts for all random choices but if x not in L then for every oracle y the machine rejects with high probability. Arora and Safra (1992) characterized NP as PCP(log n, (loglogn)/sup O(1)/). The authors improve on their result by showing that NP=PCP(logn, 1). The result has the following consequences: (1) MAXSNP-hard problems (e.g. metric TSP, MAX-SAT, MAX-CUT) 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.<<ETX>>

The Space Complexity of Approximating the Frequency Moments

The frequency moments of a sequence containingmielements of typei, 1?i?n, are the numbersFk=?ni=1mki. We consider the space complexity of randomized algorithms that approximate the numbersFk, when the elements of the sequence are given one by one and cannot be stored. Surprisingly, it turns out that the numbersF0,F1, andF2can be approximated in logarithmic space, whereas the approximation ofFkfork?6 requiresn?(1)space. Applications to data bases are mentioned as well.

Checking computations in polylogarithmic time

Motivated by Manuel Blum’s concept of inst ante checking, we consider new, very fast and generic mechanisms of checking computations. Our results exploit recent advances in interactive proof protocols [LFKN], [Sh], and especially the MIP = NEXP protocol from [BFL]. 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: (i) the same instances remain accepted; (ii) each instance/witness pair becomes checkable in poly!ogariihmic Monte Carlo time; and (iii) a witness satisfying S’ can be computed in polynomial time from a witness satisfying S. Here the instance and the description of S have to be provided in error-correcting code (since the checker will not notice slight changes). A modification of the MIP proof was required to achieve polynomial time in (iii); the earlier technique yields N“(*OglOg’1 time only. This result becomes significant if software and hardware reliability are regarded aa a considerable cost factor. The polylogarithmic checker is the only part of the system that needs to be trusted; it can be hard wired. (We use just one Checker for all problems!) The checker is tiny and so presumably can be optimized and checked off-line at a modest cost. In this setup, a single reliable PC can monitor the operation of a herd of supercomputers working with possibly extremely powerful but unreliable software and untested hardware. 1Research partially supported by NSF Grant CCR-871OO78. Email: laci@cs.uchicago. edu 2Rese~& partiay supported by NSF Grant CCR-SMXI

Interactive proofs and the hardness of approximating cliques

XEK5. E-mail: fortnow@cs.uc&cago. edu 3Supported by NSF grant CCR-SJC115276. E-mail: Lnd@cs.bu.edu 4111 cummington St., Boston MA 02215. 5E-mail: mario@cs.uchicago .edu ~1100 E 58th St, Chicago IL 60637. Permission to copy without fee all or part of this msterisl is granted provided that tie copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee snd/or specific permission. @ 1991 ACM 089791-397-31911000410021

Approximating clique is almost NP-complete

1.50 Leonid A. Levin 3

On the degree of Boolean functions as real polynomials

The contribution of this paper is two-fold. First, a connection is established between approximating the size of the largest clique in a graph and multi-prover interactive proofs. Second, an efficient multi-prover interactive proof for NP languages is constructed, where the verifier uses very few random bits and communication bits. Last, 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. Of independent interest is our proof of correctness for the multilinearity test of functions.

Quantum speed-up of Markov chain based algorithms

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, then EXPTIME=NEXPTIME and NP=P.<<ETX>>

Multiparty protocols, pseudorandom generators for logspace, and time-space trade-offs

Every Boolean function may be represented as a real polynomial. In this paper, we characterize the degree of this polynomial in terms of certain combinatorial properties of the Boolean function.Our first result is a tight lower bound of Ω(logn) on the degree needed to represent any Boolean function that depends onn variables.Our second result states that for every Boolean functionf, the following measures are all polynomially related:o The decision tree complexity off.o The degree of the polynomial representingf.o The smallest degree of a polynomialapproximating f in theLmax norm.