2 research outputs found
Partition bound is quadratically tight for product distributions
Let be a 2-party
function. For every product distribution on ,
we show that
where is the distributional communication
complexity of with error at most under the distribution
and is the {\em partition bound} of , as defined by
Jain and Klauck [{\em Proc. 25th CCC}, 2010]. We also prove a similar bound in
terms of , the {\em information complexity} of ,
namely, The latter bound was recently and
independently established by Kol [{\em Proc. 48th STOC}, 2016] using a
different technique.
We show a similar result for query complexity under product distributions.
Let be a function. For every bit-wise
product distribution on , we show that
where
is the distributional query complexity of
with error at most under the distribution and
is the {\em query partition bound} of the function
.
Partition bounds were introduced (in both communication complexity and query
complexity models) to provide LP-based lower bounds for randomized
communication complexity and randomized query complexity. Our results
demonstrate that these lower bounds are polynomially tight for {\em product}
distributions.Comment: The previous version of the paper erroneously stated the main result
in terms of relaxed partition number instead of partition numbe
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