11,864 research outputs found
Lower Bounds for Symbolic Computation on Graphs: Strongly Connected Components, Liveness, Safety, and Diameter
A model of computation that is widely used in the formal analysis of reactive
systems is symbolic algorithms. In this model the access to the input graph is
restricted to consist of symbolic operations, which are expensive in comparison
to the standard RAM operations. We give lower bounds on the number of symbolic
operations for basic graph problems such as the computation of the strongly
connected components and of the approximate diameter as well as for fundamental
problems in model checking such as safety, liveness, and co-liveness. Our lower
bounds are linear in the number of vertices of the graph, even for
constant-diameter graphs. For none of these problems lower bounds on the number
of symbolic operations were known before. The lower bounds show an interesting
separation of these problems from the reachability problem, which can be solved
with symbolic operations, where is the diameter of the graph.
Additionally we present an approximation algorithm for the graph diameter
which requires symbolic steps to achieve a
-approximation for any constant . This compares to
symbolic steps for the (naive) exact algorithm and
symbolic steps for a 2-approximation. Finally we also give a refined analysis
of the strongly connected components algorithms of Gentilini et al., showing
that it uses an optimal number of symbolic steps that is proportional to the
sum of the diameters of the strongly connected components
An -Time Algorithm for Computing Maximum Independent Set in Graphs with Bounded Degree 3
We give an -time, polynomial space algorithm for computing
Maximum Independent Set in graphs with bounded degree 3. This improves all the
previous running time bounds known for the problem
Kernelization and Sparseness: the case of Dominating Set
We prove that for every positive integer and for every graph class
of bounded expansion, the -Dominating Set problem admits a
linear kernel on graphs from . Moreover, when is only
assumed to be nowhere dense, then we give an almost linear kernel on for the classic Dominating Set problem, i.e., for the case . These
results generalize a line of previous research on finding linear kernels for
Dominating Set and -Dominating Set. However, the approach taken in this
work, which is based on the theory of sparse graphs, is radically different and
conceptually much simpler than the previous approaches.
We complement our findings by showing that for the closely related Connected
Dominating Set problem, the existence of such kernelization algorithms is
unlikely, even though the problem is known to admit a linear kernel on
-topological-minor-free graphs. Also, we prove that for any somewhere dense
class , there is some for which -Dominating Set is
W[]-hard on . Thus, our results fall short of proving a sharp
dichotomy for the parameterized complexity of -Dominating Set on
subgraph-monotone graph classes: we conjecture that the border of tractability
lies exactly between nowhere dense and somewhere dense graph classes.Comment: v2: new author, added results for r-Dominating Sets in bounded
expansion graph
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