67 research outputs found
Mining Patterns in Networks using Homomorphism
In recent years many algorithms have been developed for finding patterns in
graphs and networks. A disadvantage of these algorithms is that they use
subgraph isomorphism to determine the support of a graph pattern; subgraph
isomorphism is a well-known NP complete problem. In this paper, we propose an
alternative approach which mines tree patterns in networks by using subgraph
homomorphism. The advantage of homomorphism is that it can be computed in
polynomial time, which allows us to develop an algorithm that mines tree
patterns in arbitrary graphs in incremental polynomial time. Homomorphism
however entails two problems not found when using isomorphism: (1) two patterns
of different size can be equivalent; (2) patterns of unbounded size can be
frequent. In this paper we formalize these problems and study solutions that
easily fit within our algorithm
Counting Subgraphs in Somewhere Dense Graphs
We study the problems of counting copies and induced copies of a small
pattern graph in a large host graph . Recent work fully classified the
complexity of those problems according to structural restrictions on the
patterns . In this work, we address the more challenging task of analysing
the complexity for restricted patterns and restricted hosts. Specifically we
ask which families of allowed patterns and hosts imply fixed-parameter
tractability, i.e., the existence of an algorithm running in time for some computable function . Our main results present
exhaustive and explicit complexity classifications for families that satisfy
natural closure properties. Among others, we identify the problems of counting
small matchings and independent sets in subgraph-closed graph classes
as our central objects of study and establish the following crisp
dichotomies as consequences of the Exponential Time Hypothesis: (1) Counting
-matchings in a graph is fixed-parameter tractable if and
only if is nowhere dense. (2) Counting -independent sets in a
graph is fixed-parameter tractable if and only if
is nowhere dense. Moreover, we obtain almost tight conditional
lower bounds if is somewhere dense, i.e., not nowhere dense.
These base cases of our classifications subsume a wide variety of previous
results on the matching and independent set problem, such as counting
-matchings in bipartite graphs (Curticapean, Marx; FOCS 14), in
-colourable graphs (Roth, Wellnitz; SODA 20), and in degenerate graphs
(Bressan, Roth; FOCS 21), as well as counting -independent sets in bipartite
graphs (Curticapean et al.; Algorithmica 19).Comment: 35 pages, 3 figures, 4 tables, abstract shortened due to ArXiv
requirement
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