9 research outputs found
On the Enumeration of Minimal Dominating Sets and Related Notions
A dominating set in a graph is a subset of its vertex set such that each
vertex is either in or has a neighbour in . In this paper, we are
interested in the enumeration of (inclusion-wise) minimal dominating sets in
graphs, called the Dom-Enum problem. It is well known that this problem can be
polynomially reduced to the Trans-Enum problem in hypergraphs, i.e., the
problem of enumerating all minimal transversals in a hypergraph. Firstly we
show that the Trans-Enum problem can be polynomially reduced to the Dom-Enum
problem. As a consequence there exists an output-polynomial time algorithm for
the Trans-Enum problem if and only if there exists one for the Dom-Enum
problem. Secondly, we study the Dom-Enum problem in some graph classes. We give
an output-polynomial time algorithm for the Dom-Enum problem in split graphs,
and introduce the completion of a graph to obtain an output-polynomial time
algorithm for the Dom-Enum problem in -free chordal graphs, a proper
superclass of split graphs. Finally, we investigate the complexity of the
enumeration of (inclusion-wise) minimal connected dominating sets and minimal
total dominating sets of graphs. We show that there exists an output-polynomial
time algorithm for the Dom-Enum problem (or equivalently Trans-Enum problem) if
and only if there exists one for the following enumeration problems: minimal
total dominating sets, minimal total dominating sets in split graphs, minimal
connected dominating sets in split graphs, minimal dominating sets in
co-bipartite graphs.Comment: 15 pages, 3 figures, In revisio
Efficient Enumerations for Minimal Multicuts and Multiway Cuts
Let be an undirected graph and let be a
set of terminal pairs. A node/edge multicut is a subset of vertices/edges of
whose removal destroys all the paths between every terminal pair in .
The problem of computing a {\em minimum} node/edge multicut is NP-hard and
extensively studied from several viewpoints. In this paper, we study the
problem of enumerating all {\em minimal} node multicuts. We give an incremental
polynomial delay enumeration algorithm for minimal node multicuts, which
extends an enumeration algorithm due to Khachiyan et al. (Algorithmica, 2008)
for minimal edge multicuts. Important special cases of node/edge multicuts are
node/edge {\em multiway cuts}, where the set of terminal pairs contains every
pair of vertices in some subset , that is, . We
improve the running time bound for this special case: We devise a polynomial
delay and exponential space enumeration algorithm for minimal node multiway
cuts and a polynomial delay and space enumeration algorithm for minimal edge
multiway cuts
Approximately counting locally-optimal structures
A locally-optimal structure is a combinatorial structure such as a maximal
independent set that cannot be improved by certain (greedy) local moves, even
though it may not be globally optimal. It is trivial to construct an
independent set in a graph. It is easy to (greedily) construct a maximal
independent set. However, it is NP-hard to construct a globally-optimal
(maximum) independent set. In general, constructing a locally-optimal structure
is somewhat more difficult than constructing an arbitrary structure, and
constructing a globally-optimal structure is more difficult than constructing a
locally-optimal structure. The same situation arises with listing. The
differences between the problems become obscured when we move from listing to
counting because nearly everything is #P-complete. However, we highlight an
interesting phenomenon that arises in approximate counting, where the situation
is apparently reversed. Specifically, we show that counting maximal independent
sets is complete for #P with respect to approximation-preserving reductions,
whereas counting all independent sets, or counting maximum independent sets is
complete for an apparently smaller class, which has a
prominent role in the complexity of approximate counting. Motivated by the
difficulty of approximately counting maximal independent sets in bipartite
graphs, we also study the problem of approximately counting other
locally-optimal structures that arise in algorithmic applications, particularly
problems involving minimal separators and minimal edge separators. Minimal
separators have applications via fixed-parameter-tractable algorithms for
constructing triangulations and phylogenetic trees. Although exact
(exponential-time) algorithms exist for listing these structures, we show that
the counting problems are #P-complete with respect to both exact and
approximation-preserving reductions.Comment: Accepted to JCSS, preliminary version accepted to ICALP 2015 (Track
A
Properties, Learning Algorithms, and Applications of Chain Graphs and Bayesian Hypergraphs
Probabilistic graphical models (PGMs) use graphs, either undirected, directed, or mixed, to represent possible dependencies among the variables of a multivariate probability distri- bution. PGMs, such as Bayesian networks and Markov networks, are now widely accepted as a powerful and mature framework for reasoning and decision making under uncertainty in knowledge-based systems. With the increase of their popularity, the range of graphical models being investigated and used has also expanded. Several types of graphs with dif- ferent conditional independence interpretations - also known as Markov properties - have been proposed and used in graphical models.
The graphical structure of a Bayesian network has the form of a directed acyclic graph (DAG), which has the advantage of supporting an interpretation of the graph in terms of cause-effect relationships. However, a limitation is that only asymmetric relationships, such as cause and effect relationships, can be modeled between variables in a DAG. Chain graphs, which admit both directed and undirected edges, can be used to overcome this limitation. Today there exist three main different interpretations of chain graphs in the lit- erature. These are the Lauritzen-Wermuth-Frydenberg, the Andersson-Madigan-Perlman, and the multivariate regression interpretations. In this thesis, we study these interpreta- tions based on their separation criteria and the intuition behind their edges. Since structure learning is a critical component in constructing an intelligent system based on a chain graph model, we propose new feasible and efficient structure learning algorithms to learn chain graphs from data under the faithfulness assumption.
The proliferation of different PGMs that allow factorizations of different kinds leads us to consider a more general graphical structure in this thesis, namely directed acyclic hypergraphs. Directed acyclic hypergraphs are the graphical structure of a new proba- bilistic graphical model that we call Bayesian hypergraphs. Since there are many more hypergraphs than DAGs, undirected graphs, chain graphs, and, indeed, other graph-based networks, Bayesian hypergraphs can model much finer factorizations and thus are more computationally efficient. Bayesian hypergraphs also allow a modeler to represent causal patterns of interaction such as Noisy-OR graphically (without additional annotations). We introduce global, local and pairwise Markov properties of Bayesian hypergraphs and prove under which conditions they are equivalent. We also extend the causal interpretation of LWF chain graphs to Bayesian hypergraphs and provide corresponding formulas and a graphical criterion for intervention.
The framework of graphical models, which provides algorithms for discovering and analyzing structure in complex distributions to describe them succinctly and extract un- structured information, allows them to be constructed and utilized effectively. Two of the most important applications of graphical models are causal inference and information ex- traction. To address these abilities of graphical models, we conduct a causal analysis, comparing the performance behavior of highly-configurable systems across environmen- tal conditions (changing workload, hardware, and software versions), to explore when and how causal knowledge can be commonly exploited for performance analysis