20,508 research outputs found
The Power of an Example: Hidden Set Size Approximation Using Group Queries and Conditional Sampling
We study a basic problem of approximating the size of an unknown set in a
known universe . We consider two versions of the problem. In both versions
the algorithm can specify subsets . In the first version, which
we refer to as the group query or subset query version, the algorithm is told
whether is non-empty. In the second version, which we refer to as the
subset sampling version, if is non-empty, then the algorithm receives
a uniformly selected element from . We study the difference between
these two versions under different conditions on the subsets that the algorithm
may query/sample, and in both the case that the algorithm is adaptive and the
case where it is non-adaptive. In particular we focus on a natural family of
allowed subsets, which correspond to intervals, as well as variants of this
family
On The Communication Complexity of Linear Algebraic Problems in the Message Passing Model
We study the communication complexity of linear algebraic problems over
finite fields in the multi-player message passing model, proving a number of
tight lower bounds. Specifically, for a matrix which is distributed among a
number of players, we consider the problem of determining its rank, of
computing entries in its inverse, and of solving linear equations. We also
consider related problems such as computing the generalized inner product of
vectors held on different servers. We give a general framework for reducing
these multi-player problems to their two-player counterparts, showing that the
randomized -player communication complexity of these problems is at least
times the randomized two-player communication complexity. Provided the
problem has a certain amount of algebraic symmetry, which we formally define,
we can show the hardest input distribution is a symmetric distribution, and
therefore apply a recent multi-player lower bound technique of Phillips et al.
Further, we give new two-player lower bounds for a number of these problems. In
particular, our optimal lower bound for the two-player version of the matrix
rank problem resolves an open question of Sun and Wang.
A common feature of our lower bounds is that they apply even to the special
"threshold promise" versions of these problems, wherein the underlying
quantity, e.g., rank, is promised to be one of just two values, one on each
side of some critical threshold. These kinds of promise problems are
commonplace in the literature on data streaming as sources of hardness for
reductions giving space lower bounds
Sublinear-Time Algorithms for Monomer-Dimer Systems on Bounded Degree Graphs
For a graph , let be the partition function of the
monomer-dimer system defined by , where is the
number of matchings of size in . We consider graphs of bounded degree
and develop a sublinear-time algorithm for estimating at an
arbitrary value within additive error with high
probability. The query complexity of our algorithm does not depend on the size
of and is polynomial in , and we also provide a lower bound
quadratic in for this problem. This is the first analysis of a
sublinear-time approximation algorithm for a # P-complete problem. Our
approach is based on the correlation decay of the Gibbs distribution associated
with . We show that our algorithm approximates the probability
for a vertex to be covered by a matching, sampled according to this Gibbs
distribution, in a near-optimal sublinear time. We extend our results to
approximate the average size and the entropy of such a matching within an
additive error with high probability, where again the query complexity is
polynomial in and the lower bound is quadratic in .
Our algorithms are simple to implement and of practical use when dealing with
massive datasets. Our results extend to other systems where the correlation
decay is known to hold as for the independent set problem up to the critical
activity
Approximation Limits of Linear Programs (Beyond Hierarchies)
We develop a framework for approximation limits of polynomial-size linear
programs from lower bounds on the nonnegative ranks of suitably defined
matrices. This framework yields unconditional impossibility results that are
applicable to any linear program as opposed to only programs generated by
hierarchies. Using our framework, we prove that O(n^{1/2-eps})-approximations
for CLIQUE require linear programs of size 2^{n^\Omega(eps)}. (This lower bound
applies to linear programs using a certain encoding of CLIQUE as a linear
optimization problem.) Moreover, we establish a similar result for
approximations of semidefinite programs by linear programs. Our main ingredient
is a quantitative improvement of Razborov's rectangle corruption lemma for the
high error regime, which gives strong lower bounds on the nonnegative rank of
certain perturbations of the unique disjointness matrix.Comment: 23 pages, 2 figure
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