3 research outputs found
On Quadratic Programming with a Ratio Objective
Quadratic Programming (QP) is the well-studied problem of maximizing over
{-1,1} values the quadratic form \sum_{i \ne j} a_{ij} x_i x_j. QP captures
many known combinatorial optimization problems, and assuming the unique games
conjecture, semidefinite programming techniques give optimal approximation
algorithms. We extend this body of work by initiating the study of Quadratic
Programming problems where the variables take values in the domain {-1,0,1}.
The specific problems we study are
QP-Ratio : \max_{\{-1,0,1\}^n} \frac{\sum_{i \not = j} a_{ij} x_i x_j}{\sum
x_i^2}, and Normalized QP-Ratio : \max_{\{-1,0,1\}^n} \frac{\sum_{i \not = j}
a_{ij} x_i x_j}{\sum d_i x_i^2}, where d_i = \sum_j |a_{ij}|
We consider an SDP relaxation obtained by adding constraints to the natural
eigenvalue (or SDP) relaxation for this problem. Using this, we obtain an
algorithm for QP-ratio. We also obtain an
approximation for bipartite graphs, and better algorithms
for special cases. As with other problems with ratio objectives (e.g. uniform
sparsest cut), it seems difficult to obtain inapproximability results based on
P!=NP. We give two results that indicate that QP-Ratio is hard to approximate
to within any constant factor. We also give a natural distribution on instances
of QP-Ratio for which an n^\epsilon approximation (for \epsilon roughly 1/10)
seems out of reach of current techniques
Testing microscopic discretization
What can we say about the spectra of a collection of microscopic variables
when only their coarse-grained sums are experimentally accessible? In this
paper, using the tools and methodology from the study of quantum nonlocality,
we develop a mathematical theory of the macroscopic fluctuations generated by
ensembles of independent microscopic discrete systems. We provide algorithms to
decide which multivariate gaussian distributions can be approximated by sums of
finitely-valued random vectors. We study non-trivial cases where the
microscopic variables have an unbounded range, as well as asymptotic scenarios
with infinitely many macroscopic variables. From a foundational point of view,
our results imply that bipartite gaussian states of light cannot be understood
as beams of independent d-dimensional particle pairs. It is also shown that the
classical description of certain macroscopic optical experiments, as opposed to
the quantum one, requires variables with infinite cardinality spectra.Comment: Proof of strong NP-hardness. Connection with random walks. New
asymptotic results. Numerous typos correcte
Sum of squares lower bounds for refuting any CSP
Let be a nontrivial -ary predicate. Consider a
random instance of the constraint satisfaction problem on
variables with constraints, each being applied to randomly
chosen literals. Provided the constraint density satisfies , such
an instance is unsatisfiable with high probability. The \emph{refutation}
problem is to efficiently find a proof of unsatisfiability.
We show that whenever the predicate supports a -\emph{wise uniform}
probability distribution on its satisfying assignments, the sum of squares
(SOS) algorithm of degree
(which runs in time ) \emph{cannot} refute a random instance of
. In particular, the polynomial-time SOS algorithm requires
constraints to refute random instances of
CSP when supports a -wise uniform distribution on its satisfying
assignments. Together with recent work of Lee et al. [LRS15], our result also
implies that \emph{any} polynomial-size semidefinite programming relaxation for
refutation requires at least constraints.
Our results (which also extend with no change to CSPs over larger alphabets)
subsume all previously known lower bounds for semialgebraic refutation of
random CSPs. For every constraint predicate~, they give a three-way hardness
tradeoff between the density of constraints, the SOS degree (hence running
time), and the strength of the refutation. By recent algorithmic results of
Allen et al. [AOW15] and Raghavendra et al. [RRS16], this full three-way
tradeoff is \emph{tight}, up to lower-order factors.Comment: 39 pages, 1 figur