László Babai

Non-deterministic exponential time has two-prover interactive protocols

We determine the exact power of two-prover interactive proof systems introduced by Ben-Or, Goldwasser, Kilian, and Wigderson (1988). In this system, two all-powerful noncommunicating provers convince a randomizing polynomial time verifier in polynomial time that the inputx belongs to the languageL. We show that the class of languages having tow-prover interactive proof systems is nondeterministic exponential time. We also show that to prove membership in languages inEXP, the honest provers need the power ofEXP only. The first part of the proof of the main result extends recent techniques of polynomial extrapolation used in the single prover case by Lund, Fortnow, Karloff, Nisan, and Shamir. The second part is averification scheme for multilinearity of a function in several variables held by an oracle and can be viewed as an independent result onprogram verification. Its proof rests on combinatorial techniques employing a simple isoperimetric inequality for certain graphs:

A fast and simple randomized parallel algorithm for the maximal independent set problem

Abstract A simple parallel randomized algorithm to find a maximal independent set in a graph G = ( V , E ) on n vertices is presented. Its expected running time on a concurrent-read concurrent-write PRAM with O (| E | d max ) processors is O (log n ), where d max denotes the maximum degree. On an exclusive-read exclusive-write PRAM with O (| E |) processors the algorithm runs in O (log 2 n ). Previously, an O (log 4 n ) deterministic algorithm was given by Karp and Wigderson for the EREW-PRAM model. This was recently (independently of our work) improved to O (log 2 n ) by M. Luby. In both cases randomized algorithms depending on pairwise independent choices were turned into deterministic algorithms. We comment on how randomized combinatorial algorithms whose analysis only depends on d -wise rather than fully independent random choices (for some constant d ) can be converted into deterministic algorithms. We apply a technique due to A. Joffe (1974) and obtain deterministic construction in fast parallel time of various combinatorial objects whose existence follows from probabilistic arguments.

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

BPP has subexponential time simulations unless EXPTIME has publishable proofs

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

Canonical labeling of graphs

1.50 Leonid A. Levin 3

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

AbstractWe show thatBPP can be simulated in subexponential time for infinitely many input lengths unless exponential timeℴ collapses to the second level of the polynomial-time hierarchy.ℴ has polynomial-size circuits andℴ has publishable proofs (EXPTIME=MA). We also show thatBPP is contained in subexponential time unless exponential time has publishable proofs for infinitely many input lengths. In addition, we showBPP can be simulated in subexponential time for infinitely many input lengths unless there exist unary languages inMA-P.The proofs are based on the recent characterization of the power of multiprover interactive protocols and on random self-reducibility via low-degree polynomials. They exhibit an interplay between Boolean circuit simulation, interactive proofs and classical complexity classes. An important feature of this proof is that it does not relativize.One of the ingredients of our proof is a lemma that states that ifEXPTIME has polynomial size circuits thenEXPTIME=MA. This extends previous work by Albert Meyer.

On The Complexity Of Matrix Group Problems I

We announce an algebraic approach to the problem of assigning <italic>canonical forms</italic> to graphs. We compute canonical forms and the associated canonical labelings (or renumberings) in polynomial time for graphs of bounded valence, in moderately exponential, exp(n<supscrpt>½ + &ogr;(1)</supscrpt>),time for general graphs, in subexponential, n<supscrpt>log n</supscrpt>, time for tournaments and for 2-(&ngr;,&kgr;,λ) block designs with &kgr;,λ bounded and n<supscrpt>log log n</supscrpt> time for λ-planes (symmetric designs) with λ bounded. We prove some related problems NP-hard and indicate some open problems.

Random Graph Isomorphism

Let f(x1, …, xk) be a Boolean function that k parties wish to collaboratively evaluate, where each xi is a bit-string of length n. The ith party knows each input argument except xi; and each party has unlimited computational power. They share a blackboard, viewed by all parties, where they can exchange messages. The objective is to minimize the number of bits written on the board. We prove lower bounds of the form Ω(n · c−k), for the number of bits that need to be exchanged in order to compute some (explicitly given) polynomial time computable functions. Our bounds hold even if the parties only wish to have a 1 % advantage at guessing the value of f on random inputs. The lower bound proofs are based on discrepancy upper bounds for specific functions over “cylinder intersection” sets. These results may be of independent interest. We give several applications of the lower bounds. The first application is a pseudorandom generator for Logspace. We explicitly construct (in polynomial time pseudorandom sequences of length n from a random seed of length exp(c √log n) that no Logspace Turing machine will be able to distinguish from truly random sequences. As a corollary we give an explicit construction of a universal traversal sequence of length exp(exp(c√log n)) for arbitrary undirected graphs on n vertices. We then apply the multiparty protocol lower bounds to derive several new time-space trade-offs. We give a tight time-space trade-off of the form TS =Θ(n2), for general, k-head Turing machines; the bounds hold for a function that can be computed in linear time and constant space by a k + 1-head Turing machine. We also give a new length-width trade-off for oblivious branching programs; in particular, our bound implies new lower bounds on the size of arbitrary branching programs, or on the size of Boolean formulas (over an arbitrary finite base). Using universal hashing, Nisan has recently constructed considerably improved random generators for Logspace, with the implication of shorter explicit universal traversal sequences. The time-space and related trade-off results mentioned above are not affected by this development.

Local expansion of vertex-transitive graphs and random generation in finite groups

We build a theory of black box groups, and apply it to matrix groups over finite fields. Elements of a black box group are encoded by strings of uniform length and group operations are performd by an oracle. Subgroups are given by a list of generators. We prove that for such subgroups, membership and divisor of the order are in NPB. (B is the group box oracle.) Under a plausible mathematical hypothesis on short presentations of finite simple groups, nom membership and exaact order will also be in NPB and thus in NPB ∩ NPB.

Nondeterministic exponential time has two-prover interactive protocols

A straightforward linear time canonical labeling algorithm is shown to apply to almost all graphs (i.e. all but