4,111 research outputs found
Digraph Complexity Measures and Applications in Formal Language Theory
We investigate structural complexity measures on digraphs, in particular the
cycle rank. This concept is intimately related to a classical topic in formal
language theory, namely the star height of regular languages. We explore this
connection, and obtain several new algorithmic insights regarding both cycle
rank and star height. Among other results, we show that computing the cycle
rank is NP-complete, even for sparse digraphs of maximum outdegree 2.
Notwithstanding, we provide both a polynomial-time approximation algorithm and
an exponential-time exact algorithm for this problem. The former algorithm
yields an O((log n)^(3/2))- approximation in polynomial time, whereas the
latter yields the optimum solution, and runs in time and space O*(1.9129^n) on
digraphs of maximum outdegree at most two. Regarding the star height problem,
we identify a subclass of the regular languages for which we can precisely
determine the computational complexity of the star height problem. Namely, the
star height problem for bideterministic languages is NP-complete, and this
holds already for binary alphabets. Then we translate the algorithmic results
concerning cycle rank to the bideterministic star height problem, thus giving a
polynomial-time approximation as well as a reasonably fast exact exponential
algorithm for bideterministic star height.Comment: 19 pages, 1 figur
Learning Topic Models and Latent Bayesian Networks Under Expansion Constraints
Unsupervised estimation of latent variable models is a fundamental problem
central to numerous applications of machine learning and statistics. This work
presents a principled approach for estimating broad classes of such models,
including probabilistic topic models and latent linear Bayesian networks, using
only second-order observed moments. The sufficient conditions for
identifiability of these models are primarily based on weak expansion
constraints on the topic-word matrix, for topic models, and on the directed
acyclic graph, for Bayesian networks. Because no assumptions are made on the
distribution among the latent variables, the approach can handle arbitrary
correlations among the topics or latent factors. In addition, a tractable
learning method via optimization is proposed and studied in numerical
experiments.Comment: 38 pages, 6 figures, 2 tables, applications in topic models and
Bayesian networks are studied. Simulation section is adde
On semi-transitive orientability of Kneser graphs and their complements
An orientation of a graph is semi-transitive if it is acyclic, and for any
directed path either
there is no edge between and , or is an edge
for all . An undirected graph is semi-transitive if it admits
a semi-transitive orientation. Semi-transitive graphs include several important
classes of graphs such as 3-colorable graphs, comparability graphs, and circle
graphs, and they are precisely the class of word-representable graphs studied
extensively in the literature.
In this paper, we study semi-transitive orientability of the celebrated
Kneser graph , which is the graph whose vertices correspond to the
-element subsets of a set of elements, and where two vertices are
adjacent if and only if the two corresponding sets are disjoint. We show that
for , is not semi-transitive, while for , is semi-transitive. Also, we show computationally that a
subgraph on 16 vertices and 36 edges of , and thus itself
on 56 vertices and 280 edges, is non-semi-transitive. and are the
first explicit examples of triangle-free non-semi-transitive graphs, whose
existence was established via Erd\H{o}s' theorem by Halld\'{o}rsson et al. in
2011. Moreover, we show that the complement graph of
is semi-transitive if and only if
Geometry of the faithfulness assumption in causal inference
Many algorithms for inferring causality rely heavily on the faithfulness
assumption. The main justification for imposing this assumption is that the set
of unfaithful distributions has Lebesgue measure zero, since it can be seen as
a collection of hypersurfaces in a hypercube. However, due to sampling error
the faithfulness condition alone is not sufficient for statistical estimation,
and strong-faithfulness has been proposed and assumed to achieve uniform or
high-dimensional consistency. In contrast to the plain faithfulness assumption,
the set of distributions that is not strong-faithful has nonzero Lebesgue
measure and in fact, can be surprisingly large as we show in this paper. We
study the strong-faithfulness condition from a geometric and combinatorial
point of view and give upper and lower bounds on the Lebesgue measure of
strong-faithful distributions for various classes of directed acyclic graphs.
Our results imply fundamental limitations for the PC-algorithm and potentially
also for other algorithms based on partial correlation testing in the Gaussian
case.Comment: Published in at http://dx.doi.org/10.1214/12-AOS1080 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
"Graph Entropy, Network Coding and Guessing games"
We introduce the (private) entropy of a directed graph (in a new network coding sense) as well as a number of related concepts. We show that the entropy of a directed graph is identical to its guessing number and can be bounded from below with the number of vertices minus the size of the graph’s shortest index code. We show that the Network Coding solvability of each specific multiple unicast network is completely determined by the entropy (as well as by the shortest index code) of the directed graph that occur by identifying each source node with each corresponding target node. Shannon’s information inequalities can be used to calculate up- per bounds on a graph’s entropy as well as calculating the size of the minimal index code. Recently, a number of new families of so-called non-shannon-type information inequalities have been discovered. It has been shown that there exist communication networks with a ca- pacity strictly ess than required for solvability, but where this fact cannot be derived using Shannon’s classical information inequalities. Based on this result we show that there exist graphs with an entropy that cannot be calculated using only Shannon’s classical information inequalities, and show that better estimate can be obtained by use of certain non-shannon-type information inequalities
Homotopy Type of the Boolean Complex of a Coxeter System
In any Coxeter group, the set of elements whose principal order ideals are
boolean forms a simplicial poset under the Bruhat order. This simplicial poset
defines a cell complex, called the boolean complex. In this paper it is shown
that, for any Coxeter system of rank n, the boolean complex is homotopy
equivalent to a wedge of (n-1)-dimensional spheres. The number of such spheres
can be computed recursively from the unlabeled Coxeter graph, and defines a new
graph invariant called the boolean number. Specific calculations of the boolean
number are given for all finite and affine irreducible Coxeter systems, as well
as for systems with graphs that are disconnected, complete, or stars. One
implication of these results is that the boolean complex is contractible if and
only if a generator of the Coxeter system is in the center of the group. of
these results is that the boolean complex is contractible if and only if a
generator of the Coxeter system is in the center of the group.Comment: final version, to appear in Advances in Mathematic
A Transition-Based Directed Acyclic Graph Parser for UCCA
We present the first parser for UCCA, a cross-linguistically applicable
framework for semantic representation, which builds on extensive typological
work and supports rapid annotation. UCCA poses a challenge for existing parsing
techniques, as it exhibits reentrancy (resulting in DAG structures),
discontinuous structures and non-terminal nodes corresponding to complex
semantic units. To our knowledge, the conjunction of these formal properties is
not supported by any existing parser. Our transition-based parser, which uses a
novel transition set and features based on bidirectional LSTMs, has value not
just for UCCA parsing: its ability to handle more general graph structures can
inform the development of parsers for other semantic DAG structures, and in
languages that frequently use discontinuous structures.Comment: 16 pages; Accepted as long paper at ACL201
- …