21 research outputs found
Complete Problems for Multi-Pseudodeterministic Computations
We exhibit several computational problems that are complete for multi-pseudodeterministic computations in the following sense: (1) these problems admit 2-pseudodeterministic algorithms (2) if there exists a pseudodeterministic algorithm for any of these problems, then any multi-valued function that admits a k-pseudodeterministic algorithm for a constant k, also admits a pseudodeterministic algorithm. We also show that these computational problems are complete for Search-BPP: a pseudodeterministic algorithm for any of these problems implies a pseudodeterministic algorithm for all problems in Search-BPP
Pseudodeterministic constructions in subexponential time
We study pseudodeterministic constructions, i.e., randomized algorithms which output the same solution on most computation paths. We establish unconditionally that there is an infinite sequence {pn}n∈N of increasing primes and a randomized algorithm A running in expected sub-exponential time such that for each n, on input 1|pn|, A outputs pn with probability 1. In other words, our result provides a pseudodeterministic construction of primes in sub-exponential time which works infinitely often.
This result follows from a much more general theorem about pseudodeterministic constructions. A property Q ⊆ {0, 1}* is γ-dense if for large enough n, |Q ⋂ {0, 1}n| ≥ γ2n. We show that for each c > 0 at least one of the following holds: (1) There is a pseudodeterministic polynomial time construction of a family {Hn} of sets, Hn ⊆ {0, 1}n, such that for each (1=nc)-dense property Q ∈ DTIME(n^c) and every large enough n, Hn ⋂ Q ≠ ∅; or (2) There is a deterministic sub-exponential time construction of a family {H'n} of sets, H'n ⊆ {0, 1}n, such that for each (1/n^c)-dense property Q ∈ DTIME(n^c) and for infinitely many values of n, H'n ⋂ Q ≠ ∅.
We provide further algorithmic applications that might be of independent interest. Perhaps intriguingly, while our main results are unconditional, they have a non-constructive element, arising from a sequence of applications of the hardness versus randomness paradigm.</p
Geometry of Rounding: Near Optimal Bounds and a New Neighborhood Sperner's Lemma
A partition of is called a
-secluded partition if, for every ,
the ball intersects at most
members of . A goal in designing such secluded partitions is to
minimize while making as large as possible. This partition
problem has connections to a diverse range of topics, including deterministic
rounding schemes, pseudodeterminism, replicability, as well as Sperner/KKM-type
results.
In this work, we establish near-optimal relationships between and
. We show that, for any bounded measure partitions and for any
, it must be that . Thus, when is
restricted to , it follows that . This bound is tight up to log factors, as it is
known that there exist secluded partitions with and
. We also provide new constructions of secluded
partitions that work for a broad spectrum of and
parameters. Specifically, we prove that, for any
, there is a secluded partition with
and
. These new partitions are optimal up to
factors for various choices of and . Based
on the lower bound result, we establish a new neighborhood version of Sperner's
lemma over hypercubes, which is of independent interest. In addition, we prove
a no-free-lunch theorem about the limitations of rounding schemes in the
context of pseudodeterministic/replicable algorithms
Polynomial-Time Pseudodeterministic Construction of Primes
A randomized algorithm for a search problem is *pseudodeterministic* if it
produces a fixed canonical solution to the search problem with high
probability. In their seminal work on the topic, Gat and Goldwasser posed as
their main open problem whether prime numbers can be pseudodeterministically
constructed in polynomial time.
We provide a positive solution to this question in the infinitely-often
regime. In more detail, we give an *unconditional* polynomial-time randomized
algorithm such that, for infinitely many values of , outputs a
canonical -bit prime with high probability. More generally, we prove
that for every dense property of strings that can be decided in polynomial
time, there is an infinitely-often pseudodeterministic polynomial-time
construction of strings satisfying . This improves upon a
subexponential-time construction of Oliveira and Santhanam.
Our construction uses several new ideas, including a novel bootstrapping
technique for pseudodeterministic constructions, and a quantitative
optimization of the uniform hardness-randomness framework of Chen and Tell,
using a variant of the Shaltiel--Umans generator
On the Pseudo-Deterministic Query Complexity of NP Search Problems
We study pseudo-deterministic query complexity - randomized query algorithms that are required to output the same answer with high probability on all inputs. We prove Ω(√n) lower bounds on the pseudo-deterministic complexity of a large family of search problems based on unsatisfiable random CNF instances, and also for the promise problem (FIND1) of finding a 1 in a vector populated with at least half one’s. This gives an exponential separation between randomized query complexity and pseudo-deterministic complexity, which is tight in the quantum setting. As applications we partially solve a related combinatorial coloring problem, and we separate random tree-like Resolution from its pseudo-deterministic version. In contrast to our lower bound, we show, surprisingly, that in the zero-error, average case setting, the three notions (deterministic, randomized, pseudo-deterministic) collapse
Geometry of Rounding
Rounding has proven to be a fundamental tool in theoretical computer science.
By observing that rounding and partitioning of are equivalent,
we introduce the following natural partition problem which we call the {\em
secluded hypercube partition problem}: Given (ideally small)
and (ideally large), is there a partition of with
unit hypercubes such that for every point , its closed
-neighborhood (in the norm) intersects at most
hypercubes?
We undertake a comprehensive study of this partition problem. We prove that
for every , there is an explicit (and efficiently computable)
hypercube partition of with and . We complement this construction by proving that the value of
is the best possible (for any ) for a broad class of
``reasonable'' partitions including hypercube partitions. We also investigate
the optimality of the parameter and prove that any partition in this
broad class that has , must have .
These bounds imply limitations of certain deterministic rounding schemes
existing in the literature. Furthermore, this general bound is based on the
currently known lower bounds for the dissection number of the cube, and
improvements to this bound will yield improvements to our bounds.
While our work is motivated by the desire to understand rounding algorithms,
one of our main conceptual contributions is the introduction of the {\em
secluded hypercube partition problem}, which fits well with a long history of
investigations by mathematicians on various hypercube partitions/tilings of
Euclidean space
Complexity Theory
Computational Complexity Theory is the mathematical study of the intrinsic power and limitations of computational resources like time, space, or randomness. The current workshop focused on recent developments in various sub-areas including arithmetic complexity, Boolean complexity, communication complexity, cryptography, probabilistic proof systems, pseudorandomness, and quantum computation. Many of the developments are related to diverse mathematical fields such as algebraic geometry, combinatorial number theory, probability theory, representation theory, and the theory of error-correcting codes