333 research outputs found
Quadratically Tight Relations for Randomized Query Complexity
Let be a Boolean function. The certificate
complexity is a complexity measure that is quadratically tight for the
zero-error randomized query complexity : . In this paper we study a new complexity measure that we call
expectational certificate complexity , which is also a quadratically
tight bound on : . We prove that and show that there is a quadratic separation between
the two, thus gives a tighter upper bound for . The measure is
also related to the fractional certificate complexity as follows:
. This also connects to an open question by
Aaronson whether is a quadratically tight bound for , as
is in fact a relaxation of .
In the second part of the work, we upper bound the distributed query
complexity for product distributions by the square of
the query corruption bound () which improves upon a
result of Harsha, Jain and Radhakrishnan [2015]. A similar statement for
communication complexity is open.Comment: 14 page
The Partition Bound for Classical Communication Complexity and Query Complexity
We describe new lower bounds for randomized communication complexity and
query complexity which we call the partition bounds. They are expressed as the
optimum value of linear programs. For communication complexity we show that the
partition bound is stronger than both the rectangle/corruption bound and the
\gamma_2/generalized discrepancy bounds. In the model of query complexity we
show that the partition bound is stronger than the approximate polynomial
degree and classical adversary bounds. We also exhibit an example where the
partition bound is quadratically larger than polynomial degree and classical
adversary bounds.Comment: 28 pages, ver. 2, added conten
Low-Sensitivity Functions from Unambiguous Certificates
We provide new query complexity separations against sensitivity for total
Boolean functions: a power separation between deterministic (and even
randomized or quantum) query complexity and sensitivity, and a power
separation between certificate complexity and sensitivity. We get these
separations by using a new connection between sensitivity and a seemingly
unrelated measure called one-sided unambiguous certificate complexity
(). We also show that is lower-bounded by fractional block
sensitivity, which means we cannot use these techniques to get a
super-quadratic separation between and . We also provide a
quadratic separation between the tree-sensitivity and decision tree complexity
of Boolean functions, disproving a conjecture of Gopalan, Servedio, Tal, and
Wigderson (CCC 2016).
Along the way, we give a power separation between certificate
complexity and one-sided unambiguous certificate complexity, improving the
power separation due to G\"o\"os (FOCS 2015). As a consequence, we
obtain an improved lower-bound on the
co-nondeterministic communication complexity of the Clique vs. Independent Set
problem.Comment: 25 pages. This version expands the results and adds Pooya Hatami and
Avishay Tal as author
Separating decision tree complexity from subcube partition complexity
The subcube partition model of computation is at least as powerful as
decision trees but no separation between these models was known. We show that
there exists a function whose deterministic subcube partition complexity is
asymptotically smaller than its randomized decision tree complexity, resolving
an open problem of Friedgut, Kahn, and Wigderson (2002). Our lower bound is
based on the information-theoretic techniques first introduced to lower bound
the randomized decision tree complexity of the recursive majority function.
We also show that the public-coin partition bound, the best known lower bound
method for randomized decision tree complexity subsuming other general
techniques such as block sensitivity, approximate degree, randomized
certificate complexity, and the classical adversary bound, also lower bounds
randomized subcube partition complexity. This shows that all these lower bound
techniques cannot prove optimal lower bounds for randomized decision tree
complexity, which answers an open question of Jain and Klauck (2010) and Jain,
Lee, and Vishnoi (2014).Comment: 16 pages, 1 figur
Weak Parity
We study the query complexity of Weak Parity: the problem of computing the
parity of an n-bit input string, where one only has to succeed on a 1/2+eps
fraction of input strings, but must do so with high probability on those inputs
where one does succeed. It is well-known that n randomized queries and n/2
quantum queries are needed to compute parity on all inputs. But surprisingly,
we give a randomized algorithm for Weak Parity that makes only
O(n/log^0.246(1/eps)) queries, as well as a quantum algorithm that makes only
O(n/sqrt(log(1/eps))) queries. We also prove a lower bound of
Omega(n/log(1/eps)) in both cases; and using extremal combinatorics, prove
lower bounds of Omega(log n) in the randomized case and Omega(sqrt(log n)) in
the quantum case for any eps>0. We show that improving our lower bounds is
intimately related to two longstanding open problems about Boolean functions:
the Sensitivity Conjecture, and the relationships between query complexity and
polynomial degree.Comment: 18 page
Certificate games
We introduce and study Certificate Game complexity, a measure of complexity
based on the probability of winning a game where two players are given inputs
with different function values and are asked to output such that (zero-communication setting).
We give upper and lower bounds for private coin, public coin, shared
entanglement and non-signaling strategies, and give some separations. We show
that complexity in the public coin model is upper bounded by Randomized query
and Certificate complexity. On the other hand, it is lower bounded by
fractional and randomized certificate complexity, making it a good candidate to
prove strong lower bounds on randomized query complexity. Complexity in the
private coin model is bounded from below by zero-error randomized query
complexity.
The quantum measure highlights an interesting and surprising difference
between classical and quantum query models. Whereas the public coin certificate
game complexity is bounded from above by randomized query complexity, the
quantum certificate game complexity can be quadratically larger than quantum
query complexity. We use non-signaling, a notion from quantum information, to
give a lower bound of on the quantum certificate game complexity of the
function, whose quantum query complexity is , then go on
to show that this ``non-signaling bottleneck'' applies to all functions with
high sensitivity, block sensitivity or fractional block sensitivity.
We consider the single-bit version of certificate games (inputs of the two
players have Hamming distance ). We prove that the single-bit version of
certificate game complexity with shared randomness is equal to sensitivity up
to constant factors, giving a new characterization of sensitivity. The
single-bit version with private randomness is equal to , where
is the spectral sensitivity.Comment: 43 pages, 1 figure, ITCS202
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