122 research outputs found
Structural Average Case Complexity
AbstractLevin introduced an average-case complexity measure, based on a notion of âpolynomial on average,â and defined âaverage-case polynomial-time many-one reducibilityâ among randomized decision problems. We generalize his notions of average-case complexity classes, Random-NP and Average-P. Ben-Davidet al. use the notation of ăC, Fă to denote the set of randomized decision problems (L, Ό) such thatLis a set in C andÎŒis a probability density function in F. This paper introduces AverăC, Fă as the class of randomized decision problems (L, Ό) such thatLis computed by a type-C machine onÎŒ-average andÎŒis a density function in F. These notations capture all known average-case complexity classes as, for example, Random-NP= ăNP, P-compă and Average-P=AverăP, âă, where P-comp denotes the set of density functions whose distributions are computable in polynomial time, and â denotes the set of all density functions. Mainly studied are polynomial-time reductions between randomized decision problems: manyâone, deterministic Turing and nondeterministic Turing reductions and the average-case versions of them. Based on these reducibilities, structural properties of average-case complexity classes are discussed. We give average-case analogues of concepts in worst-case complexity theory; in particular, the polynomial time hierarchy and Turing self-reducibility, and we show that all known complete sets for Random-NP are Turing self-reducible. A new notion of âreal polynomial-time computationsâ is introduced based on average polynomial-time computations for arbitrary distributions from a fixed set, and it is used to characterize the worst-case complexity classesÎpkandÎŁpkof the polynomial-time hierarchy
Restricted Information from Nonadaptive Queries to NP
AbstractWe investigate classes of sets that can be decided by bounded truth-table reductions to an NP set in which evaluators donothave full access to the answers to the queries but get only restricted information such as the number of queries that are in the oracle set or even just this number modulom, for somemâ©Ÿ2. We also investigate the case in which evaluators are nondeterministic. We show that when we vary the information that the evaluators get, this can change the resulting power of the evaluators. We locate all these classes within levels of the Boolean hierarchy which allows us to compare the complexity of such classes
Approximate solution of NP optimization problems
AbstractThis paper presents the main results obtained in the field of approximation algorithms in a unified framework. Most of these results have been revisited in order to emphasize two basic tools useful for characterizing approximation classes, that is, combinatorial properties of problems and approximation preserving reducibilities. In particular, after reviewing the most important combinatorial characterizations of the classes PTAS and FPTAS, we concentrate on the class APX and, as a concluding result, we show that this class coincides with the class of optimization problems which are reducible to the maximum satisfiability problem with respect to a polynomial-time approximation preserving reducibility
Hamming Approximation of NP Witnesses
Given a satisfiable 3-SAT formula, how hard is it to find an assignment to
the variables that has Hamming distance at most n/2 to a satisfying assignment?
More generally, consider any polynomial-time verifier for any NP-complete
language. A d(n)-Hamming-approximation algorithm for the verifier is one that,
given any member x of the language, outputs in polynomial time a string a with
Hamming distance at most d(n) to some witness w, where (x,w) is accepted by the
verifier. Previous results have shown that, if P != NP, then every NP-complete
language has a verifier for which there is no
(n/2-n^(2/3+d))-Hamming-approximation algorithm, for various constants d > 0.
Our main result is that, if P != NP, then every paddable NP-complete language
has a verifier that admits no (n/2+O(sqrt(n log n)))-Hamming-approximation
algorithm. That is, one cannot get even half the bits right. We also consider
natural verifiers for various well-known NP-complete problems. They do have
n/2-Hamming-approximation algorithms, but, if P != NP, have no
(n/2-n^epsilon)-Hamming-approximation algorithms for any constant epsilon > 0.
We show similar results for randomized algorithms
The complexity of searching implicit graphs
AbstractThe standard complexity classes of complexity theory do not allow for direct classification of most of the problems solved by heuristic search algorithms. The reason is that, almost always, these are defined in terms of implicit graphs of state or problem reduction spaces, while the standard definitions of all complexity classes are specifically tailored to explicit inputs.To allow for more precise comparisons with standard complexity classes, we introduce here a model for the analysis of algorithms on graphs given by vertex expansion procedures. It is based on previously studied concepts of âsuccinct representationâ techniques, and allows us to prove PSPACE-completeness or EXPTIME-completeness of specific, natural problems on implicit graphs, such as those solved by Aâ, AOâ, and other best-first search strategies
Two Theorems in List Decoding
We prove the following results concerning the list decoding of
error-correcting codes:
(i) We show that for \textit{any} code with a relative distance of
(over a large enough alphabet), the following result holds for \textit{random
errors}: With high probability, for a \rho\le \delta -\eps fraction of random
errors (for any \eps>0), the received word will have only the transmitted
codeword in a Hamming ball of radius around it. Thus, for random errors,
one can correct twice the number of errors uniquely correctable from worst-case
errors for any code. A variant of our result also gives a simple algorithm to
decode Reed-Solomon codes from random errors that, to the best of our
knowledge, runs faster than known algorithms for certain ranges of parameters.
(ii) We show that concatenated codes can achieve the list decoding capacity
for erasures. A similar result for worst-case errors was proven by Guruswami
and Rudra (SODA 08), although their result does not directly imply our result.
Our results show that a subset of the random ensemble of codes considered by
Guruswami and Rudra also achieve the list decoding capacity for erasures.
Our proofs employ simple counting and probabilistic arguments.Comment: 19 pages, 0 figure
Approximating solution structure of the Weighted Sentence Alignment problem
We study the complexity of approximating solution structure of the bijective
weighted sentence alignment problem of DeNero and Klein (2008). In particular,
we consider the complexity of finding an alignment that has a significant
overlap with an optimal alignment. We discuss ways of representing the solution
for the general weighted sentence alignment as well as phrases-to-words
alignment problem, and show that computing a string which agrees with the
optimal sentence partition on more than half (plus an arbitrarily small
polynomial fraction) positions for the phrases-to-words alignment is NP-hard.
For the general weighted sentence alignment we obtain such bound from the
agreement on a little over 2/3 of the bits. Additionally, we generalize the
Hamming distance approximation of a solution structure to approximating it with
respect to the edit distance metric, obtaining similar lower bounds
A PCP Characterization of AM
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
Total Representations
Almost all representations considered in computable analysis are partial. We
provide arguments in favor of total representations (by elements of the Baire
space). Total representations make the well known analogy between numberings
and representations closer, unify some terminology, simplify some technical
details, suggest interesting open questions and new invariants of topological
spaces relevant to computable analysis.Comment: 30 page
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