272 research outputs found
Reversible MCMC on Markov equivalence classes of sparse directed acyclic graphs
Graphical models are popular statistical tools which are used to represent
dependent or causal complex systems. Statistically equivalent causal or
directed graphical models are said to belong to a Markov equivalent class. It
is of great interest to describe and understand the space of such classes.
However, with currently known algorithms, sampling over such classes is only
feasible for graphs with fewer than approximately 20 vertices. In this paper,
we design reversible irreducible Markov chains on the space of Markov
equivalent classes by proposing a perfect set of operators that determine the
transitions of the Markov chain. The stationary distribution of a proposed
Markov chain has a closed form and can be computed easily. Specifically, we
construct a concrete perfect set of operators on sparse Markov equivalence
classes by introducing appropriate conditions on each possible operator.
Algorithms and their accelerated versions are provided to efficiently generate
Markov chains and to explore properties of Markov equivalence classes of sparse
directed acyclic graphs (DAGs) with thousands of vertices. We find
experimentally that in most Markov equivalence classes of sparse DAGs, (1) most
edges are directed, (2) most undirected subgraphs are small and (3) the number
of these undirected subgraphs grows approximately linearly with the number of
vertices. The article contains supplement arXiv:1303.0632,
http://dx.doi.org/10.1214/13-AOS1125SUPPComment: Published in at http://dx.doi.org/10.1214/13-AOS1125 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Efficient Sampling and Counting of Graph Structures related to Chordal Graphs
Counting problems aim to count the number of solutions for a given input, for example, counting the number of variable assignments that satisfy a Boolean formula. Sampling problems aim to produce a random object from a desired distribution, for example, producing a variable assignment drawn uniformly at random from all assignments that satisfy a Boolean formula. The problems of counting and sampling of graph structures on different types of graphs have been studied for decades for their great importance in areas like complexity theory and statistical physics. For many graph structures such as independent sets and acyclic orientations, it is widely believed that no exact or approximate (with an arbitrarily small error) polynomial-time algorithms on general graphs exist. Therefore, the research community studies various types of graphs, aiming either to design a polynomial-time counting or sampling algorithm for such inputs, or to prove a corresponding inapproximability result. Chordal graphs have been studied widely in both AI and theoretical computer science, but their study from the counting perspective has been relatively limited. Previous works showed that some graph structures can be counted in polynomial time on chordal graphs, when their counting on general graphs is provably computationally hard. The main objective of this thesis is to design and analyze counting and sampling algorithms for several well-known graph structures, including independent sets and different types of graph orientations, on chordal graphs. Our contributions can be described from two perspectives: evaluating the performances of some well-known sampling techniques, such as Markov chain Monte Carlo, on chordal graphs; and showing that the chordality does make those counting problems polynomial-time solvable
The complexity of power indices in voting games with incompatible players
We study the complexity of computing the Banzhaf index in weighted voting games with cooperation restricted by an incompatibility graph. With an existing algorithm as a starting point, we use concepts from complexity theory to show that, for some classes of incompatibility graphs, the problem can be solved efficiently, as long as the players have "small" weights. We also show that for some other class of graphs it is unlikely that we can find efficient algorithms to compute the Banzhaf index in the corresponding restricted game. Finally, we discuss the complexity of deciding whether the index of a player is non-zero
Reconfigurations of Combinatorial Problems: Graph Colouring and Hamiltonian Cycle
We explore algorithmic aspects of two known combinatorial problems, Graph Colouring and Hamiltonian Cycle, by examining properties of their solution space. One can model the set of solutions of a combinatorial problem by the solution graph , where vertices are solutions of and there is an edge between two vertices, when the two corresponding solutions satisfy an adjacency reconfiguration rule. For example, we can define the reconfiguration rule for graph colouring to be that two solutions are adjacent when they differ in colour in exactly one vertex.
The exploration of the properties of the solution graph can give rise to interesting questions. The connectivity of is the most prominent question in this research area. This is reasonable, since the main motivation for modelling combinatorial solutions as a graph is to be able to transform one into the other in a stepwise fashion, by following paths between solutions in the graph. Connectivity questions can be made binary, that is expressed as decision problems which accept a 'yes' or 'no' answer. For example, given two specific solutions, is there a path between them? Is the graph of solutions connected?
In this thesis, we first show that the diameter of the solution graph of vertex -colourings of k-colourable chordal and chordal bipartite graphs is , where and n is the number of vertices of . Then, we formulate a decision problem on the connectivity of the graph colouring solution graph, where we allow extra colours to be used in order to enforce a path between two colourings with no path between them. We give some results for general instances and we also explore what kind of graphs pose a challenge to determine the complexity of the problem for general instances. Finally, we give a linear algorithm which decides whether there is a path between two solutions of the Hamiltonian Cycle Problem for graphs of maximum degree five, and thus providing insights towards the complexity classification of the decision problem
Peregrine: A Pattern-Aware Graph Mining System
Graph mining workloads aim to extract structural properties of a graph by
exploring its subgraph structures. General purpose graph mining systems provide
a generic runtime to explore subgraph structures of interest with the help of
user-defined functions that guide the overall exploration process. However, the
state-of-the-art graph mining systems remain largely oblivious to the shape (or
pattern) of the subgraphs that they mine. This causes them to: (a) explore
unnecessary subgraphs; (b) perform expensive computations on the explored
subgraphs; and, (c) hold intermediate partial subgraphs in memory; all of which
affect their overall performance. Furthermore, their programming models are
often tied to their underlying exploration strategies, which makes it difficult
for domain users to express complex mining tasks.
In this paper, we develop Peregrine, a pattern-aware graph mining system that
directly explores the subgraphs of interest while avoiding exploration of
unnecessary subgraphs, and simultaneously bypassing expensive computations
throughout the mining process. We design a pattern-based programming model that
treats "graph patterns" as first class constructs and enables Peregrine to
extract the semantics of patterns, which it uses to guide its exploration. Our
evaluation shows that Peregrine outperforms state-of-the-art distributed and
single machine graph mining systems, and scales to complex mining tasks on
larger graphs, while retaining simplicity and expressivity with its
"pattern-first" programming approach.Comment: This is the full version of the paper appearing in the European
Conference on Computer Systems (EuroSys), 202
LIPIcs, Volume 274, ESA 2023, Complete Volume
LIPIcs, Volume 274, ESA 2023, Complete Volum
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