436 research outputs found
Counting hypergraph matchings up to uniqueness threshold
We study the problem of approximately counting matchings in hypergraphs of
bounded maximum degree and maximum size of hyperedges. With an activity
parameter , each matching is assigned a weight .
The counting problem is formulated as computing a partition function that gives
the sum of the weights of all matchings in a hypergraph. This problem unifies
two extensively studied statistical physics models in approximate counting: the
hardcore model (graph independent sets) and the monomer-dimer model (graph
matchings).
For this model, the critical activity
is the threshold for the uniqueness of Gibbs measures on the infinite
-uniform -regular hypertree. Consider hypergraphs of maximum
degree at most and maximum size of hyperedges at most . We show that
when , there is an FPTAS for computing the partition
function; and when , there is a PTAS for computing the
log-partition function. These algorithms are based on the decay of correlation
(strong spatial mixing) property of Gibbs distributions. When , there is no PRAS for the partition function or the log-partition
function unless NPRP.
Towards obtaining a sharp transition of computational complexity of
approximate counting, we study the local convergence from a sequence of finite
hypergraphs to the infinite lattice with specified symmetry. We show a
surprising connection between the local convergence and the reversibility of a
natural random walk. This leads us to a barrier for the hardness result: The
non-uniqueness of infinite Gibbs measure is not realizable by any finite
gadgets
Most primitive groups are full automorphism groups of edge-transitive hypergraphs
We prove that, for a primitive permutation group G acting on a set of size n,
other than the alternating group, the probability that Aut(X,Y^G) = G for a
random subset Y of X, tends to 1 as n tends to infinity. So the property of the
title holds for all primitive groups except the alternating groups and finitely
many others. This answers a question of M. Klin. Moreover, we give an upper
bound n^{1/2+\epsilon} for the minimum size of the edges in such a hypergraph.
This is essentially best possible.Comment: To appear in special issue of Journal of Algebra in memory of Akos
Seres
Asymptotics for incidence matrix classes
We define {\em incidence matrices} to be zero-one matrices with no zero rows
or columns. A classification of incidence matrices is considered for which
conditions of symmetry by transposition, having no repeated rows/columns, or
identification by permutation of rows/columns are imposed. We find asymptotics
and relationships for the number of matrices with ones in these classes as
.Comment: updated and slightly expanded versio
Generalized Kneser coloring theorems with combinatorial proofs
The Kneser conjecture (1955) was proved by Lov\'asz (1978) using the
Borsuk-Ulam theorem; all subsequent proofs, extensions and generalizations also
relied on Algebraic Topology results, namely the Borsuk-Ulam theorem and its
extensions. Only in 2000, Matou\v{s}ek provided the first combinatorial proof
of the Kneser conjecture.
Here we provide a hypergraph coloring theorem, with a combinatorial proof,
which has as special cases the Kneser conjecture as well as its extensions and
generalization by (hyper)graph coloring theorems of Dol'nikov,
Alon-Frankl-Lov\'asz, Sarkaria, and Kriz. We also give a combinatorial proof of
Schrijver's theorem.Comment: 19 pages, 4 figure
Asymptotic enumeration of incidence matrices
We discuss the problem of counting {\em incidence matrices}, i.e. zero-one
matrices with no zero rows or columns. Using different approaches we give three
different proofs for the leading asymptotics for the number of matrices with
ones as . We also give refined results for the asymptotic
number of incidence matrices with ones.Comment: jpconf style files. Presented at the conference "Counting Complexity:
An international workshop on statistical mechanics and combinatorics." In
celebration of Prof. Tony Guttmann's 60th birthda
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