103 research outputs found
On the Usefulness of Predicates
Motivated by the pervasiveness of strong inapproximability results for
Max-CSPs, we introduce a relaxed notion of an approximate solution of a
Max-CSP. In this relaxed version, loosely speaking, the algorithm is allowed to
replace the constraints of an instance by some other (possibly real-valued)
constraints, and then only needs to satisfy as many of the new constraints as
possible.
To be more precise, we introduce the following notion of a predicate
being \emph{useful} for a (real-valued) objective : given an almost
satisfiable Max- instance, there is an algorithm that beats a random
assignment on the corresponding Max- instance applied to the same sets of
literals. The standard notion of a nontrivial approximation algorithm for a
Max-CSP with predicate is exactly the same as saying that is useful for
itself.
We say that is useless if it is not useful for any . This turns out to
be equivalent to the following pseudo-randomness property: given an almost
satisfiable instance of Max- it is hard to find an assignment such that the
induced distribution on -bit strings defined by the instance is not
essentially uniform.
Under the Unique Games Conjecture, we give a complete and simple
characterization of useful Max-CSPs defined by a predicate: such a Max-CSP is
useless if and only if there is a pairwise independent distribution supported
on the satisfying assignments of the predicate. It is natural to also consider
the case when no negations are allowed in the CSP instance, and we derive a
similar complete characterization (under the UGC) there as well.
Finally, we also include some results and examples shedding additional light
on the approximability of certain Max-CSPs
On the Power of Many One-Bit Provers
We study the class of languages, denoted by \MIP[k, 1-\epsilon, s], which
have -prover games where each prover just sends a \emph{single} bit, with
completeness and soundness error . For the case that
(i.e., for the case of interactive proofs), Goldreich, Vadhan and Wigderson
({\em Computational Complexity'02}) demonstrate that \SZK exactly
characterizes languages having 1-bit proof systems with"non-trivial" soundness
(i.e., ). We demonstrate that for the case that
, 1-bit -prover games exhibit a significantly richer structure:
+ (Folklore) When , \MIP[k, 1-\epsilon, s]
= \BPP;
+ When , \MIP[k,
1-\epsilon, s] = \SZK;
+ When , \AM \subseteq \MIP[k, 1-\epsilon,
s];
+ For and sufficiently large , \MIP[k, 1-\epsilon, s]
\subseteq \EXP;
+ For , \MIP[k, 1, 1-\epsilon, s] = \NEXP.
As such, 1-bit -prover games yield a natural "quantitative" approach to
relating complexity classes such as \BPP,\SZK,\AM, \EXP, and \NEXP.
We leave open the question of whether a more fine-grained hierarchy (between
\AM and \NEXP) can be established for the case when
Intermediate problems in modular circuits satisfiability
In arXiv:1710.08163 a generalization of Boolean circuits to arbitrary finite
algebras had been introduced and applied to sketch P versus NP-complete
borderline for circuits satisfiability over algebras from congruence modular
varieties. However the problem for nilpotent (which had not been shown to be
NP-hard) but not supernilpotent algebras (which had been shown to be polynomial
time) remained open.
In this paper we provide a broad class of examples, lying in this grey area,
and show that, under the Exponential Time Hypothesis and Strong Exponential
Size Hypothesis (saying that Boolean circuits need exponentially many modular
counting gates to produce boolean conjunctions of any arity), satisfiability
over these algebras have intermediate complexity between and , where measures how much a nilpotent algebra
fails to be supernilpotent. We also sketch how these examples could be used as
paradigms to fill the nilpotent versus supernilpotent gap in general.
Our examples are striking in view of the natural strong connections between
circuits satisfiability and Constraint Satisfaction Problem for which the
dichotomy had been shown by Bulatov and Zhuk
Quantum algorithms for computing short discrete logarithms and factoring RSA integers
In this paper we generalize the quantum algorithm for computing short discrete logarithms previously introduced by Ekerå so as to allow for various tradeoffs between the number of times that the algorithm need be executed on the one hand, and the complexity of the algorithm and the requirements it imposes on the quantum computer on the other hand.
Furthermore, we describe applications of algorithms for computing short discrete logarithms. In particular, we show how other important problems such as those of factoring RSA integers and of finding the order of groups under side information may be recast as short discrete logarithm problems. This immediately gives rise to an algorithm for factoring RSA integers that is less complex than Shor’s general factoring algorithm in the sense that it imposes smaller requirements on the quantum computer.
In both our algorithm and Shor’s algorithm, the main hurdle is to compute a modular exponentiation in superposition. When factoring an n bit integer, the exponent is of length 2n bits in Shor’s algorithm, compared to slightly more than n/2 bits in our algorithm
Circuit Bottom Fan-in and Computational Power
We investigate the relationship between circuit bottom fan-in and circuit size when circuit depth is xed. We show that in order to compute certain functions, a moderate reduction in circuit bottom fan-in will cause signi cant increase in circuit size. In particular, we prove that there are functions that are computable by circuits of linear size and depth k with bottom fan-in 2 but require exponential size for circuits of depth k with bottom fan-in 1. A general scheme is established to study the trade-o between circuit bottom fan-in and circuit size. Based on this scheme, we are able to prove, for example, that for any integer c, there are functions that are computable by circuits of linear size and depth k with bottom fan-in O(log n) but require exponential size for circuits of depth k with bottom fan-in c, and that for any constant> 0, there are functions that are computable by circuits of linear size and depth k with bottom fan-in log n but require superpolynomial size for circuits of depth k with bottom fan-in O(log 1; n). A consequence of these results is that the three input read-modes of alternating Turing machines proposed in the literature are all distinct
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