73,338 research outputs found
A note on the complexity of the causal ordering problem
In this note we provide a concise report on the complexity of the causal ordering problem, originally introduced by Simon to reason about causal dependencies implicit in systems of mathematical equations. We show that Simon’s classical algorithm to infer causal ordering is NP-Hard—an intractability previously guessed but never proven. We present then a detailed account based on Nayak’s suggested algorithmic solution (the best available), which is dominated by computing transitive closure—bounded in time by O(|V|·|S|), where S(E, V) is the input system structure composed of a set E of equations over a set V of variables with number of variable appearances (density) |S|. We also comment on the potential of causal ordering for emerging applications in large-scale hypothesis management and analytics
Structure and Complexity in Planning with Unary Operators
Unary operator domains -- i.e., domains in which operators have a single
effect -- arise naturally in many control problems. In its most general form,
the problem of STRIPS planning in unary operator domains is known to be as hard
as the general STRIPS planning problem -- both are PSPACE-complete. However,
unary operator domains induce a natural structure, called the domain's causal
graph. This graph relates between the preconditions and effect of each domain
operator. Causal graphs were exploited by Williams and Nayak in order to
analyze plan generation for one of the controllers in NASA's Deep-Space One
spacecraft. There, they utilized the fact that when this graph is acyclic, a
serialization ordering over any subgoal can be obtained quickly. In this paper
we conduct a comprehensive study of the relationship between the structure of a
domain's causal graph and the complexity of planning in this domain. On the
positive side, we show that a non-trivial polynomial time plan generation
algorithm exists for domains whose causal graph induces a polytree with a
constant bound on its node indegree. On the negative side, we show that even
plan existence is hard when the graph is a directed-path singly connected DAG.
More generally, we show that the number of paths in the causal graph is closely
related to the complexity of planning in the associated domain. Finally we
relate our results to the question of complexity of planning with serializable
subgoals
Inferring Regulatory Networks by Combining Perturbation Screens and Steady State Gene Expression Profiles
Reconstructing transcriptional regulatory networks is an important task in
functional genomics. Data obtained from experiments that perturb genes by
knockouts or RNA interference contain useful information for addressing this
reconstruction problem. However, such data can be limited in size and/or are
expensive to acquire. On the other hand, observational data of the organism in
steady state (e.g. wild-type) are more readily available, but their
informational content is inadequate for the task at hand. We develop a
computational approach to appropriately utilize both data sources for
estimating a regulatory network. The proposed approach is based on a three-step
algorithm to estimate the underlying directed but cyclic network, that uses as
input both perturbation screens and steady state gene expression data. In the
first step, the algorithm determines causal orderings of the genes that are
consistent with the perturbation data, by combining an exhaustive search method
with a fast heuristic that in turn couples a Monte Carlo technique with a fast
search algorithm. In the second step, for each obtained causal ordering, a
regulatory network is estimated using a penalized likelihood based method,
while in the third step a consensus network is constructed from the highest
scored ones. Extensive computational experiments show that the algorithm
performs well in reconstructing the underlying network and clearly outperforms
competing approaches that rely only on a single data source. Further, it is
established that the algorithm produces a consistent estimate of the regulatory
network.Comment: 24 pages, 4 figures, 6 table
Penalized Likelihood Methods for Estimation of Sparse High Dimensional Directed Acyclic Graphs
Directed acyclic graphs (DAGs) are commonly used to represent causal
relationships among random variables in graphical models. Applications of these
models arise in the study of physical, as well as biological systems, where
directed edges between nodes represent the influence of components of the
system on each other. The general problem of estimating DAGs from observed data
is computationally NP-hard, Moreover two directed graphs may be observationally
equivalent. When the nodes exhibit a natural ordering, the problem of
estimating directed graphs reduces to the problem of estimating the structure
of the network. In this paper, we propose a penalized likelihood approach that
directly estimates the adjacency matrix of DAGs. Both lasso and adaptive lasso
penalties are considered and an efficient algorithm is proposed for estimation
of high dimensional DAGs. We study variable selection consistency of the two
penalties when the number of variables grows to infinity with the sample size.
We show that although lasso can only consistently estimate the true network
under stringent assumptions, adaptive lasso achieves this task under mild
regularity conditions. The performance of the proposed methods is compared to
alternative methods in simulated, as well as real, data examples.Comment: 19 pages, 8 figure
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