8 research outputs found
Polynomial bounds for decoupling, with applications
Let f(x) = f(x_1, ..., x_n) = \sum_{|S| <= k} a_S \prod_{i \in S} x_i be an
n-variate real multilinear polynomial of degree at most k, where S \subseteq
[n] = {1, 2, ..., n}. For its "one-block decoupled" version,
f~(y,z) = \sum_{|S| <= k} a_S \sum_{i \in S} y_i \prod_{j \in S\i} z_j,
we show tail-bound comparisons of the form
Pr[|f~(y,z)| > C_k t] t].
Our constants C_k, D_k are significantly better than those known for "full
decoupling". For example, when x, y, z are independent Gaussians we obtain C_k
= D_k = O(k); when x, y, z, Rademacher random variables we obtain C_k = O(k^2),
D_k = k^{O(k)}. By contrast, for full decoupling only C_k = D_k = k^{O(k)} is
known in these settings.
We describe consequences of these results for query complexity (related to
conjectures of Aaronson and Ambainis) and for analysis of Boolean functions
(including an optimal sharpening of the DFKO Inequality).Comment: 19 pages, including bibliograph
Lift-and-Round to Improve Weighted Completion Time on Unrelated Machines
We consider the problem of scheduling jobs on unrelated machines so as to
minimize the sum of weighted completion times. Our main result is a
-approximation algorithm for some fixed , improving upon the
long-standing bound of 3/2 (independently due to Skutella, Journal of the ACM,
2001, and Sethuraman & Squillante, SODA, 1999). To do this, we first introduce
a new lift-and-project based SDP relaxation for the problem. This is necessary
as the previous convex programming relaxations have an integrality gap of
. Second, we give a new general bipartite-rounding procedure that produces
an assignment with certain strong negative correlation properties.Comment: 21 pages, 4 figure
The Price of Independence in an Echo Chamber with Dependence Ambiguity
How much should we pay to remove the interdependence of biased information sources? This question is relevant in both statistics and political economy. When there are many information sources or variables, their dependence may be unknown, which creates multivariate ambiguity. One approach to answer our leading question involves use of decoupling inequalities from probability theory. We present a new inequality, designed to cope with this question, which holds for any type of dependence across information sources. We apply our method to a simple formalization of a political echo chamber. For a given set of marginal information, this bound is the sup over all possible joint distributions connecting the marginals. Our method highlights a price to pay for facing summed dependent (multivariate) data, similar to the probability premium required for univariate data. We show that a conservative decisionmaker will pay approximately 50% more than if the data were independent, in order to freely neglect the correlations
Approximating Generalized Network Design under (Dis)economies of Scale with Applications to Energy Efficiency
In a generalized network design (GND) problem, a set of resources are
assigned to multiple communication requests. Each request contributes its
weight to the resources it uses and the total load on a resource is then
translated to the cost it incurs via a resource specific cost function. For
example, a request may be to establish a virtual circuit, thus contributing to
the load on each edge in the circuit. Motivated by energy efficiency
applications, recently, there is a growing interest in GND using cost functions
that exhibit (dis)economies of scale ((D)oS), namely, cost functions that
appear subadditive for small loads and superadditive for larger loads.
The current paper advances the existing literature on approximation
algorithms for GND problems with (D)oS cost functions in various aspects: (1)
we present a generic approximation framework that yields approximation results
for a much wider family of requests in both directed and undirected graphs; (2)
our framework allows for unrelated weights, thus providing the first
non-trivial approximation for the problem of scheduling unrelated parallel
machines with (D)oS cost functions; (3) our framework is fully combinatorial
and runs in strongly polynomial time; (4) the family of (D)oS cost functions
considered in the current paper is more general than the one considered in the
existing literature, providing a more accurate abstraction for practical energy
conservation scenarios; and (5) we obtain the first approximation ratio for GND
with (D)oS cost functions that depends only on the parameters of the resources'
technology and does not grow with the number of resources, the number of
requests, or their weights. The design of our framework relies heavily on
Roughgarden's smoothness toolbox (JACM 2015), thus demonstrating the possible
usefulness of this toolbox in the area of approximation algorithms.Comment: 39 pages, 1 figure. An extended abstract of this paper is to appear
in the 50th Annual ACM Symposium on the Theory of Computing (STOC 2018