942 research outputs found
Isoperimetric Inequalities and Supercritical Percolation on High-dimensional Graphs
It is known that many different types of finite random subgraph models
undergo quantitatively similar phase transitions around their percolation
thresholds, and the proofs of these results rely on isoperimetric properties of
the underlying host graph. Recently, the authors showed that such a phase
transition occurs in a large class of regular high-dimensional product graphs,
generalising a classic result for the hypercube.
In this paper we give new isoperimetric inequalities for such regular
high-dimensional product graphs, which generalise the well-known isoperimetric
inequality of Harper for the hypercube, and are asymptotically sharp for a wide
range of set sizes. We then use these isoperimetric properties to investigate
the structure of the giant component in supercritical percolation on
these product graphs, that is, when , where is the
degree of the product graph and is a small enough constant.
We show that typically has edge-expansion . Furthermore, we show that likely contains a linear-sized
subgraph with vertex-expansion . These
results are best possible up to the logarithmic factor in .
Using these likely expansion properties, we determine, up to small
polylogarithmic factors in , the likely diameter of as well as the
typical mixing time of a lazy random walk on . Furthermore, we show the
likely existence of a path of length .
These results not only generalise, but also improve substantially upon the
known bounds in the case of the hypercube, where in particular the likely
diameter and typical mixing time of were previously only known to be
polynomial in
The Connectivity of Boolean Satisfiability: Dichotomies for Formulas and Circuits
For Boolean satisfiability problems, the structure of the solution space is
characterized by the solution graph, where the vertices are the solutions, and
two solutions are connected iff they differ in exactly one variable. In 2006,
Gopalan et al. studied connectivity properties of the solution graph and
related complexity issues for CSPs, motivated mainly by research on
satisfiability algorithms and the satisfiability threshold. They proved
dichotomies for the diameter of connected components and for the complexity of
the st-connectivity question, and conjectured a trichotomy for the connectivity
question. Recently, we were able to establish the trichotomy [arXiv:1312.4524].
Here, we consider connectivity issues of satisfiability problems defined by
Boolean circuits and propositional formulas that use gates, resp. connectives,
from a fixed set of Boolean functions. We obtain dichotomies for the diameter
and the two connectivity problems: on one side, the diameter is linear in the
number of variables, and both problems are in P, while on the other side, the
diameter can be exponential, and the problems are PSPACE-complete. For
partially quantified formulas, we show an analogous dichotomy.Comment: 20 pages, several improvement
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