10 research outputs found
A remark on pseudo proof systems and hard instances of the satisfiability problem
We link two concepts from the literature, namely hard sequences for the satisfiability problem sat and so-called pseudo proof systems proposed for study by Krajícek. Pseudo proof systems are elements of a particular nonstandard model constructed by forcing with random variables. We show that the existence of mad pseudo proof systems is equivalent to the existence of a randomized polynomial time procedure with a highly restrictive use of randomness which produces satisfiable formulas whose satisfying assignments are probably hard to find.Peer ReviewedPostprint (published version
A New View on Worst-Case to Average-Case Reductions for NP Problems
We study the result by Bogdanov and Trevisan (FOCS, 2003), who show that
under reasonable assumptions, there is no non-adaptive worst-case to
average-case reduction that bases the average-case hardness of an NP-problem on
the worst-case complexity of an NP-complete problem. We replace the hiding and
the heavy samples protocol in [BT03] by employing the histogram verification
protocol of Haitner, Mahmoody and Xiao (CCC, 2010), which proves to be very
useful in this context. Once the histogram is verified, our hiding protocol is
directly public-coin, whereas the intuition behind the original protocol
inherently relies on private coins
Average-Case Complexity
We survey the average-case complexity of problems in NP.
We discuss various notions of good-on-average algorithms, and present
completeness results due to Impagliazzo and Levin. Such completeness results
establish the fact that if a certain specific (but somewhat artificial) NP
problem is easy-on-average with respect to the uniform distribution, then all
problems in NP are easy-on-average with respect to all samplable distributions.
Applying the theory to natural distributional problems remain an outstanding
open question. We review some natural distributional problems whose
average-case complexity is of particular interest and that do not yet fit into
this theory.
A major open question whether the existence of hard-on-average problems in NP
can be based on the PNP assumption or on related worst-case assumptions.
We review negative results showing that certain proof techniques cannot prove
such a result. While the relation between worst-case and average-case
complexity for general NP problems remains open, there has been progress in
understanding the relation between different ``degrees'' of average-case
complexity. We discuss some of these ``hardness amplification'' results