22 research outputs found
Counting independent sets in hypergraphs
Let be a triangle-free graph with vertices and average degree . We
show that contains at least independent sets. This improves a recent result of the
first and third authors \cite{countingind}. In particular, it implies that as
, every triangle-free graph on vertices has at least
independent sets, where . Further, we show that for all , there exists a triangle-free
graph with vertices which has at most
independent sets, where . This disproves a
conjecture from \cite{countingind}.
Let be a -uniform linear hypergraph with vertices and average
degree . We also show that there exists a constant such that the
number of independent sets in is at least This is tight apart from the constant
and generalizes a result of Duke, Lefmann, and R\"odl
\cite{uncrowdedrodl}, which guarantees the existence of an independent set of
size . Both of our lower bounds follow
from a more general statement, which applies to hereditary properties of
hypergraphs
The complexity of approximating bounded-degree Boolean #CSP
AbstractThe degree of a CSP instance is the maximum number of times that any variable appears in the scopes of constraints. We consider the approximate counting problem for Boolean CSP with bounded-degree instances, for constraint languages containing the two unary constant relations {0} and {1}. When the maximum allowed degree is large enough (at least 6) we obtain a complete classification of the complexity of this problem. It is exactly solvable in polynomial time if every relation in the constraint language is affine. It is equivalent to the problem of approximately counting independent sets in bipartite graphs if every relation can be expressed as conjunctions of {0}, {1} and binary implication. Otherwise, there is no FPRAS unless NP=RP. For lower degree bounds, additional cases arise, where the complexity is related to the complexity of approximately counting independent sets in hypergraphs
Positive independence densities of finite rank countable hypergraphs are achieved by finite hypergraphs
The independence density of a finite hypergraph is the probability that a
subset of vertices, chosen uniformly at random contains no hyperedges.
Independence densities can be generalized to countable hypergraphs using
limits. We show that, in fact, every positive independence density of a
countably infinite hypergraph with hyperedges of bounded size is equal to the
independence density of some finite hypergraph whose hyperedges are no larger
than those in the infinite hypergraph. This answers a question of Bonato,
Brown, Kemkes, and Pra{\l}at about independence densities of graphs.
Furthermore, we show that for any , the set of independence densities of
hypergraphs with hyperedges of size at most is closed and contains no
infinite increasing sequences.Comment: To appear in the European Journal of Combinatorics, 12 page
Path Coupling Using Stopping Times and Counting Independent Sets and Colourings in Hypergraphs
We give a new method for analysing the mixing time of a Markov chain using
path coupling with stopping times. We apply this approach to two hypergraph
problems. We show that the Glauber dynamics for independent sets in a
hypergraph mixes rapidly as long as the maximum degree Delta of a vertex and
the minimum size m of an edge satisfy m>= 2Delta+1. We also show that the
Glauber dynamics for proper q-colourings of a hypergraph mixes rapidly if m>= 4
and q > Delta, and if m=3 and q>=1.65Delta. We give related results on the
hardness of exact and approximate counting for both problems.Comment: Simpler proof of main theorem. Improved bound on mixing time. 19
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