69 research outputs found
Regularity lemmas in a Banach space setting
Szemer\'edi's regularity lemma is a fundamental tool in extremal graph
theory, theoretical computer science and combinatorial number theory. Lov\'asz
and Szegedy [L. Lov\'asz and B. Szegedy: Szemer\'edi's Lemma for the analyst,
Geometric and Functional Analysis 17 (2007), 252-270] gave a Hilbert space
interpretation of the lemma and an interpretation in terms of compact- ness of
the space of graph limits. In this paper we prove several compactness results
in a Banach space setting, generalising results of Lov\'asz and Szegedy as well
as a result of Borgs, Chayes, Cohn and Zhao [C. Borgs, J.T. Chayes, H. Cohn and
Y. Zhao: An Lp theory of sparse graph convergence I: limits, sparse random
graph models, and power law distributions, arXiv preprint arXiv:1401.2906
(2014)].Comment: 15 pages. The topological part has been substantially improved based
on referees comments. To appear in European Journal of Combinatoric
Deterministic polynomial-time approximation algorithms for partition functions and graph polynomials
In this paper we show a new way of constructing deterministic polynomial-time
approximation algorithms for computing complex-valued evaluations of a large
class of graph polynomials on bounded degree graphs. In particular, our
approach works for the Tutte polynomial and independence polynomial, as well as
partition functions of complex-valued spin and edge-coloring models.
More specifically, we define a large class of graph polynomials
and show that if and there is a disk centered at zero in the
complex plane such that does not vanish on for all bounded degree
graphs , then for each in the interior of there exists a
deterministic polynomial-time approximation algorithm for evaluating at
. This gives an explicit connection between absence of zeros of graph
polynomials and the existence of efficient approximation algorithms, allowing
us to show new relationships between well-known conjectures.
Our work builds on a recent line of work initiated by. Barvinok, which
provides a new algorithmic approach besides the existing Markov chain Monte
Carlo method and the correlation decay method for these types of problems.Comment: 27 pages; some changes have been made based on referee comments. In
particular a tiny error in Proposition 4.4 has been fixed. The introduction
and concluding remarks have also been rewritten to incorporate the most
recent developments. Accepted for publication in SIAM Journal on Computatio
Tensor invariants for certain subgroups of the orthogonal group
Let V be an n-dimensional vector space and let On be the orthogonal group.
Motivated by a question of B. Szegedy (B. Szegedy, Edge coloring models and
reflection positivity, Journal of the American Mathematical Society Volume 20,
Number 4, 2007), about the rank of edge connection matrices of partition
functions of vertex models, we give a combinatorial parameterization of tensors
in V \otimes k invariant under certain subgroups of the orthogonal group. This
allows us to give an answer to this question for vertex models with values in
an algebraically closed field of characteristic zero.Comment: 14 pages, figure. We fixed a few typo's. To appear in Journal of
Algebraic Combinatoric
Weighted counting of solutions to sparse systems of equations
Given complex numbers , we define the weight of a
set of 0-1 vectors as the sum of over all
vectors in . We present an algorithm, which for a set
defined by a system of homogeneous linear equations with at most
variables per equation and at most equations per variable, computes
within relative error in time
provided for an absolute constant and all . A similar algorithm is constructed for computing
the weight of a linear code over . Applications include counting
weighted perfect matchings in hypergraphs, counting weighted graph
homomorphisms, computing weight enumerators of linear codes with sparse code
generating matrices, and computing the partition functions of the ferromagnetic
Potts model at low temperatures and of the hard-core model at high fugacity on
biregular bipartite graphs.Comment: The exposition is improved, a couple of inaccuracies correcte
On the Caratheodory rank of polymatroid bases
In this paper we prove that the Carath\'eodory rank of the set of bases of a
(poly)matroid is upper bounded by the cardinality of the ground set.Comment: 7 page
Polyhedra with the Integer Caratheodory Property
A polyhedron P has the Integer Caratheodory Property if the following holds.
For any positive integer k and any integer vector w in kP, there exist affinely
independent integer vectors x_1,...,x_t in P and positive integers n_1,...,n_t
such that n_1+...+n_t=k and w=n_1x_1+...+n_tx_t. In this paper we prove that if
P is a (poly)matroid base polytope or if P is defined by a TU matrix, then P
and projections of P satisfy the integer Caratheodory property.Comment: 12 page
The rank of edge connection matrices and the dimension of algebras of invariant tensors
We characterize the rank of edge connection matrices of partition functions
of real vertex models, as the dimension of the homogeneous components of the
algebra of -invariant tensors. Here is the sub- group of the real
orthogonal group that stabilizes the vertex model. This answers a question of
Bal\'azs Szegedy from 2007.Comment: Two figures added and some typos fixe
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