472 research outputs found
NP-Hardness of Approximately Solving Linear Equations Over Reals
URL lists article on conference siteIn this paper, we consider the problem of approximately solving a system of homogeneous
linear equations over reals, where each equation contains at most three variables.
Since the all-zero assignment always satisfies all the equations exactly, we restrict the
assignments to be “non-trivial”. Here is an informal statement of our result: it is NP-hard
to distinguish whether there is a non-trivial assignment that satisfies fraction of the
equations or every non-trivial assignment fails to satisfy a constant fraction of the equations
with a ``margin" of .
We develop linearity and dictatorship testing procedures for functions f : Rn 7--> R over
a Gaussian space, which could be of independent interest.
We believe that studying the complexity of linear equations over reals, apart from being
a natural pursuit, can lead to progress on the Unique Games Conjecture.National Science Foundation (U.S.) (NSF CAREER grant CCF-0833228)National Science Foundation (U.S.) (Expeditions grant CCF-0832795)U.S.-Israel Binational Science Foundation (BSF grant 2008059
On the Complexity of Reconstructing Chemical Reaction Networks
The analysis of the structure of chemical reaction networks is crucial for a
better understanding of chemical processes. Such networks are well described as
hypergraphs. However, due to the available methods, analyses regarding network
properties are typically made on standard graphs derived from the full
hypergraph description, e.g.\ on the so-called species and reaction graphs.
However, a reconstruction of the underlying hypergraph from these graphs is not
necessarily unique. In this paper, we address the problem of reconstructing a
hypergraph from its species and reaction graph and show NP-completeness of the
problem in its Boolean formulation. Furthermore we study the problem
empirically on random and real world instances in order to investigate its
computational limits in practice
Computing a Nonnegative Matrix Factorization -- Provably
In the Nonnegative Matrix Factorization (NMF) problem we are given an nonnegative matrix and an integer . Our goal is to express
as where and are nonnegative matrices of size
and respectively. In some applications, it makes sense to ask
instead for the product to approximate -- i.e. (approximately)
minimize \norm{M - AW}_F where \norm{}_F denotes the Frobenius norm; we
refer to this as Approximate NMF. This problem has a rich history spanning
quantum mechanics, probability theory, data analysis, polyhedral combinatorics,
communication complexity, demography, chemometrics, etc. In the past decade NMF
has become enormously popular in machine learning, where and are
computed using a variety of local search heuristics. Vavasis proved that this
problem is NP-complete. We initiate a study of when this problem is solvable in
polynomial time:
1. We give a polynomial-time algorithm for exact and approximate NMF for
every constant . Indeed NMF is most interesting in applications precisely
when is small.
2. We complement this with a hardness result, that if exact NMF can be solved
in time , 3-SAT has a sub-exponential time algorithm. This rules
out substantial improvements to the above algorithm.
3. We give an algorithm that runs in time polynomial in , and
under the separablity condition identified by Donoho and Stodden in 2003. The
algorithm may be practical since it is simple and noise tolerant (under benign
assumptions). Separability is believed to hold in many practical settings.
To the best of our knowledge, this last result is the first example of a
polynomial-time algorithm that provably works under a non-trivial condition on
the input and we believe that this will be an interesting and important
direction for future work.Comment: 29 pages, 3 figure
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