13,943 research outputs found
Pseudorandomness for Approximate Counting and Sampling
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
NP-hardness of circuit minimization for multi-output functions
Can we design efficient algorithms for finding fast algorithms? This question is captured by various circuit minimization problems, and algorithms for the corresponding tasks have significant practical applications. Following the work of Cook and Levin in the early 1970s, a central question is whether minimizing the circuit size of an explicitly given function is NP-complete. While this is known to hold in restricted models such as DNFs, making progress with respect to more expressive classes of circuits has been elusive.
In this work, we establish the first NP-hardness result for circuit minimization of total functions in the setting of general (unrestricted) Boolean circuits. More precisely, we show that computing the minimum circuit size of a given multi-output Boolean function f : {0,1}^n ? {0,1}^m is NP-hard under many-one polynomial-time randomized reductions. Our argument builds on a simpler NP-hardness proof for the circuit minimization problem for (single-output) Boolean functions under an extended set of generators.
Complementing these results, we investigate the computational hardness of minimizing communication. We establish that several variants of this problem are NP-hard under deterministic reductions. In particular, unless ? = ??, no polynomial-time computable function can approximate the deterministic two-party communication complexity of a partial Boolean function up to a polynomial. This has consequences for the class of structural results that one might hope to show about the communication complexity of partial functions
Hardness Amplification of Optimization Problems
In this paper, we prove a general hardness amplification scheme for optimization problems based on the technique of direct products.
We say that an optimization problem ? is direct product feasible if it is possible to efficiently aggregate any k instances of ? and form one large instance of ? such that given an optimal feasible solution to the larger instance, we can efficiently find optimal feasible solutions to all the k smaller instances. Given a direct product feasible optimization problem ?, our hardness amplification theorem may be informally stated as follows:
If there is a distribution D over instances of ? of size n such that every randomized algorithm running in time t(n) fails to solve ? on 1/?(n) fraction of inputs sampled from D, then, assuming some relationships on ?(n) and t(n), there is a distribution D\u27 over instances of ? of size O(n??(n)) such that every randomized algorithm running in time t(n)/poly(?(n)) fails to solve ? on 99/100 fraction of inputs sampled from D\u27.
As a consequence of the above theorem, we show hardness amplification of problems in various classes such as NP-hard problems like Max-Clique, Knapsack, and Max-SAT, problems in P such as Longest Common Subsequence, Edit Distance, Matrix Multiplication, and even problems in TFNP such as Factoring and computing Nash equilibrium
From average case complexity to improper learning complexity
The basic problem in the PAC model of computational learning theory is to
determine which hypothesis classes are efficiently learnable. There is
presently a dearth of results showing hardness of learning problems. Moreover,
the existing lower bounds fall short of the best known algorithms.
The biggest challenge in proving complexity results is to establish hardness
of {\em improper learning} (a.k.a. representation independent learning).The
difficulty in proving lower bounds for improper learning is that the standard
reductions from -hard problems do not seem to apply in this
context. There is essentially only one known approach to proving lower bounds
on improper learning. It was initiated in (Kearns and Valiant 89) and relies on
cryptographic assumptions.
We introduce a new technique for proving hardness of improper learning, based
on reductions from problems that are hard on average. We put forward a (fairly
strong) generalization of Feige's assumption (Feige 02) about the complexity of
refuting random constraint satisfaction problems. Combining this assumption
with our new technique yields far reaching implications. In particular,
1. Learning 's is hard.
2. Agnostically learning halfspaces with a constant approximation ratio is
hard.
3. Learning an intersection of halfspaces is hard.Comment: 34 page
Sum of squares lower bounds for refuting any CSP
Let be a nontrivial -ary predicate. Consider a
random instance of the constraint satisfaction problem on
variables with constraints, each being applied to randomly
chosen literals. Provided the constraint density satisfies , such
an instance is unsatisfiable with high probability. The \emph{refutation}
problem is to efficiently find a proof of unsatisfiability.
We show that whenever the predicate supports a -\emph{wise uniform}
probability distribution on its satisfying assignments, the sum of squares
(SOS) algorithm of degree
(which runs in time ) \emph{cannot} refute a random instance of
. In particular, the polynomial-time SOS algorithm requires
constraints to refute random instances of
CSP when supports a -wise uniform distribution on its satisfying
assignments. Together with recent work of Lee et al. [LRS15], our result also
implies that \emph{any} polynomial-size semidefinite programming relaxation for
refutation requires at least constraints.
Our results (which also extend with no change to CSPs over larger alphabets)
subsume all previously known lower bounds for semialgebraic refutation of
random CSPs. For every constraint predicate~, they give a three-way hardness
tradeoff between the density of constraints, the SOS degree (hence running
time), and the strength of the refutation. By recent algorithmic results of
Allen et al. [AOW15] and Raghavendra et al. [RRS16], this full three-way
tradeoff is \emph{tight}, up to lower-order factors.Comment: 39 pages, 1 figur
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