201 research outputs found
Active Learning for Undirected Graphical Model Selection
This paper studies graphical model selection, i.e., the problem of estimating
a graph of statistical relationships among a collection of random variables.
Conventional graphical model selection algorithms are passive, i.e., they
require all the measurements to have been collected before processing begins.
We propose an active learning algorithm that uses junction tree representations
to adapt future measurements based on the information gathered from prior
measurements. We prove that, under certain conditions, our active learning
algorithm requires fewer scalar measurements than any passive algorithm to
reliably estimate a graph. A range of numerical results validate our theory and
demonstrates the benefits of active learning.Comment: AISTATS 201
Canonizing Graphs of Bounded Tree Width in Logspace
Graph canonization is the problem of computing a unique representative, a
canon, from the isomorphism class of a given graph. This implies that two
graphs are isomorphic exactly if their canons are equal. We show that graphs of
bounded tree width can be canonized by logarithmic-space (logspace) algorithms.
This implies that the isomorphism problem for graphs of bounded tree width can
be decided in logspace. In the light of isomorphism for trees being hard for
the complexity class logspace, this makes the ubiquitous class of graphs of
bounded tree width one of the few classes of graphs for which the complexity of
the isomorphism problem has been exactly determined.Comment: 26 page
Counting and Sampling from Markov Equivalent DAGs Using Clique Trees
A directed acyclic graph (DAG) is the most common graphical model for
representing causal relationships among a set of variables. When restricted to
using only observational data, the structure of the ground truth DAG is
identifiable only up to Markov equivalence, based on conditional independence
relations among the variables. Therefore, the number of DAGs equivalent to the
ground truth DAG is an indicator of the causal complexity of the underlying
structure--roughly speaking, it shows how many interventions or how much
additional information is further needed to recover the underlying DAG. In this
paper, we propose a new technique for counting the number of DAGs in a Markov
equivalence class. Our approach is based on the clique tree representation of
chordal graphs. We show that in the case of bounded degree graphs, the proposed
algorithm is polynomial time. We further demonstrate that this technique can be
utilized for uniform sampling from a Markov equivalence class, which provides a
stochastic way to enumerate DAGs in the equivalence class and may be needed for
finding the best DAG or for causal inference given the equivalence class as
input. We also extend our counting and sampling method to the case where prior
knowledge about the underlying DAG is available, and present applications of
this extension in causal experiment design and estimating the causal effect of
joint interventions
Separability and Vertex Ordering of Graphs
Many graph optimization problems, such as finding an optimal coloring, or a largest clique, can be solved by a divide-and-conquer approach. One such well-known technique is decomposition by clique separators where a graph is decomposed into special induced subgraphs along their clique separators. While the most common practice of this method employs minimal clique separators, in this work we study other variations as well. We strive to characterize their structure and in particular the bound on the number of atoms. In fact, we strengthen the known bounds for the general clique cutset decomposition and the minimal clique separator decomposition. Graph ordering is the arrangement of a graph’s vertices according to a certain logic and is a useful tool in optimization problems. Special types of vertices are often recognized in graph classes, for instance it is well-known every chordal graph contains a simplicial vertex. Vertex-ordering, based on such properties, have originated many linear time algorithms. We propose to define a new family named SE-Class such that every graph belonging to this family inherently contains a simplicial extreme, that is a vertex which is either simplicial or has exactly two neighbors which are non-adjacent. Our family lends itself to an ordering based on simplicial extreme vertices (named SEO) which we demonstrate to be advantageous for the coloring and maximum clique problems. In addition, we examine the relation of SE-Class to the family of (Even-Hole, Kite)-free graphs and show a linear time generation of SEO for (Even-Hole, Diamond, Claw)-free graphs. We showcase the applications of those two core tools, namely clique-based decomposition and vertex ordering, on the (Even-Hole, Kite)-free family
Thick Forests
We consider classes of graphs, which we call thick graphs, that have their
vertices replaced by cliques and their edges replaced by bipartite graphs. In
particular, we consider the case of thick forests, which are a subclass of
perfect graphs. We show that this class can be recognised in polynomial time,
and examine the complexity of counting independent sets and colourings for
graphs in the class. We consider some extensions of our results to thick graphs
beyond thick forests.Comment: 40 pages, 19 figure
Two New Characterizations of Path Graphs
Path graphs are intersection graphs of paths in a tree. We start from the
characterization of path graphs by Monma and Wei [C.L.~Monma,~and~V.K.~Wei,
Intersection Graphs of Paths in a Tree, J. Combin. Theory Ser. B, 41:2 (1986)
141--181] and we reduce it to some 2-colorings subproblems, obtaining the first
characterization that directly leads to a polynomial recognition algorithm.
Then we introduce the collection of the attachedness graphs of a graph and we
exhibit a list of minimal forbidden 2-edge colored subgraphs in each of the
attachedness graph.Comment: 18 pages, 6 figure
Positive-Instance Driven Dynamic Programming for Treewidth
Consider a dynamic programming scheme for a decision problem in which all
subproblems involved are also decision problems. An implementation of such a
scheme is {\em positive-instance driven} (PID), if it generates positive
subproblem instances, but not negative ones, building each on smaller positive
instances.
We take the dynamic programming scheme due to Bouchitt\'{e} and Todinca for
treewidth computation, which is based on minimal separators and potential
maximal cliques, and design a variant (for the decision version of the problem)
with a natural PID implementation. The resulting algorithm performs extremely
well: it solves a number of standard benchmark instances for which the optimal
solutions have not previously been known. Incorporating a new heuristic
algorithm for detecting safe separators, it also solves all of the 100 public
instances posed by the exact treewidth track in PACE 2017, a competition on
algorithm implementation.
We describe the algorithm, prove its correctness, and give a running time
bound in terms of the number of positive subproblem instances. We perform an
experimental analysis which supports the practical importance of such a bound.Comment: A preliminary and abridged version appeared in ESA 201
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