2,097 research outputs found
The Complexity of Bisimulation and Simulation on Finite Systems
In this paper the computational complexity of the (bi)simulation problem over
restricted graph classes is studied. For trees given as pointer structures or
terms the (bi)simulation problem is complete for logarithmic space or NC,
respectively. This solves an open problem from Balc\'azar, Gabarr\'o, and
S\'antha. Furthermore, if only one of the input graphs is required to be a
tree, the bisimulation (simulation) problem is contained in AC (LogCFL). In
contrast, it is also shown that the simulation problem is P-complete already
for graphs of bounded path-width
Performance and scalability of indexed subgraph query processing methods
Graph data management systems have become very popular
as graphs are the natural data model for many applications.
One of the main problems addressed by these systems is subgraph
query processing; i.e., given a query graph, return all
graphs that contain the query. The naive method for processing
such queries is to perform a subgraph isomorphism
test against each graph in the dataset. This obviously does
not scale, as subgraph isomorphism is NP-Complete. Thus,
many indexing methods have been proposed to reduce the
number of candidate graphs that have to underpass the subgraph
isomorphism test. In this paper, we identify a set of
key factors-parameters, that influence the performance of
related methods: namely, the number of nodes per graph,
the graph density, the number of distinct labels, the number
of graphs in the dataset, and the query graph size. We then
conduct comprehensive and systematic experiments that analyze
the sensitivity of the various methods on the values of
the key parameters. Our aims are twofold: first to derive
conclusions about the algorithms’ relative performance, and,
second, to stress-test all algorithms, deriving insights as to
their scalability, and highlight how both performance and
scalability depend on the above factors. We choose six wellestablished
indexing methods, namely Grapes, CT-Index,
GraphGrepSX, gIndex, Tree+∆, and gCode, as representative
approaches of the overall design space, including the
most recent and best performing methods. We report on
their index construction time and index size, and on query
processing performance in terms of time and false positive
ratio. We employ both real and synthetic datasets. Specifi-
cally, four real datasets of different characteristics are used:
AIDS, PDBS, PCM, and PPI. In addition, we generate a
large number of synthetic graph datasets, empowering us to
systematically study the algorithms’ performance and scalability
versus the aforementioned key parameters
Fixed-parameter tractable canonization and isomorphism test for graphs of bounded treewidth
We give a fixed-parameter tractable algorithm that, given a parameter and
two graphs , either concludes that one of these graphs has treewidth
at least , or determines whether and are isomorphic. The running
time of the algorithm on an -vertex graph is ,
and this is the first fixed-parameter algorithm for Graph Isomorphism
parameterized by treewidth.
Our algorithm in fact solves the more general canonization problem. We namely
design a procedure working in time that, for a
given graph on vertices, either concludes that the treewidth of is
at least , or: * finds in an isomorphic-invariant way a graph
that is isomorphic to ; * finds an isomorphism-invariant
construction term --- an algebraic expression that encodes together with a
tree decomposition of of width .
Hence, the isomorphism test reduces to verifying whether the computed
isomorphic copies or the construction terms for and are equal.Comment: Full version of a paper presented at FOCS 201
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