459 research outputs found

    Instance-Wise Hardness Versus Randomness Tradeoffs for Arthur-Merlin Protocols

    Get PDF

    Uniform hardness versus randomness tradeoffs for Arthur-Merlin games

    Full text link

    A PCP Characterization of AM

    Get PDF
    We introduce a 2-round stochastic constraint-satisfaction problem, and show that its approximation version is complete for (the promise version of) the complexity class AM. This gives a `PCP characterization' of AM analogous to the PCP Theorem for NP. Similar characterizations have been given for higher levels of the Polynomial Hierarchy, and for PSPACE; however, we suggest that the result for AM might be of particular significance for attempts to derandomize this class. To test this notion, we pose some `Randomized Optimization Hypotheses' related to our stochastic CSPs that (in light of our result) would imply collapse results for AM. Unfortunately, the hypotheses appear over-strong, and we present evidence against them. In the process we show that, if some language in NP is hard-on-average against circuits of size 2^{Omega(n)}, then there exist hard-on-average optimization problems of a particularly elegant form. All our proofs use a powerful form of PCPs known as Probabilistically Checkable Proofs of Proximity, and demonstrate their versatility. We also use known results on randomness-efficient soundness- and hardness-amplification. In particular, we make essential use of the Impagliazzo-Wigderson generator; our analysis relies on a recent Chernoff-type theorem for expander walks.Comment: 18 page

    Pseudorandomness for Approximate Counting and Sampling

    Get PDF
    We study computational procedures that use both randomness and nondeterminism. The goal of this paper is to derandomize such procedures under the weakest possible assumptions. Our main technical contribution allows one to “boost” a given hardness assumption: We show that if there is a problem in EXP that cannot be computed by poly-size nondeterministic circuits then there is one which cannot be computed by poly-size circuits that make non-adaptive NP oracle queries. This in particular shows that the various assumptions used over the last few years by several authors to derandomize Arthur-Merlin games (i.e., show AM = NP) are in fact all equivalent. We also define two new primitives that we regard as the natural pseudorandom objects associated with approximate counting and sampling of NP-witnesses. We use the “boosting” theorem and hashing techniques to construct these primitives using an assumption that is no stronger than that used to derandomize AM. We observe that Cai's proof that S_2^P ⊆ PP⊆(NP) and the learning algorithm of Bshouty et al. can be seen as reductions to sampling that are not probabilistic. As a consequence they can be derandomized under an assumption which is weaker than the assumption that was previously known to suffice

    AM with Multiple Merlins

    Get PDF
    We introduce and study a new model of interactive proofs: AM(k), or Arthur-Merlin with k non-communicating Merlins. Unlike with the better-known MIP, here the assumption is that each Merlin receives an independent random challenge from Arthur. One motivation for this model (which we explore in detail) comes from the close analogies between it and the quantum complexity class QMA(k), but the AM(k) model is also natural in its own right. We illustrate the power of multiple Merlins by giving an AM(2) protocol for 3SAT, in which the Merlins' challenges and responses consist of only n^{1/2+o(1)} bits each. Our protocol has the consequence that, assuming the Exponential Time Hypothesis (ETH), any algorithm for approximating a dense CSP with a polynomial-size alphabet must take n^{(log n)^{1-o(1)}} time. Algorithms nearly matching this lower bound are known, but their running times had never been previously explained. Brandao and Harrow have also recently used our 3SAT protocol to show quasipolynomial hardness for approximating the values of certain entangled games. In the other direction, we give a simple quasipolynomial-time approximation algorithm for free games, and use it to prove that, assuming the ETH, our 3SAT protocol is essentially optimal. More generally, we show that multiple Merlins never provide more than a polynomial advantage over one: that is, AM(k)=AM for all k=poly(n). The key to this result is a subsampling theorem for free games, which follows from powerful results by Alon et al. and Barak et al. on subsampling dense CSPs, and which says that the value of any free game can be closely approximated by the value of a logarithmic-sized random subgame.Comment: 48 page

    Lower Bounds and Derandomization

    Get PDF
    A major open problem in complexity theory is to determine whether randomized complexity classes such as BPP, AM, and MA have any nontrivial derandomization. This thesis investigates the derandomization of two randomized versions of the polynomial hierarchy
    • 

    corecore