43,615 research outputs found
Toric algebra of hypergraphs
The edges of any hypergraph parametrize a monomial algebra called the edge
subring of the hypergraph. We study presentation ideals of these edge subrings,
and describe their generators in terms of balanced walks on hypergraphs. Our
results generalize those for the defining ideals of edge subrings of graphs,
which are well-known in the commutative algebra community, and popular in the
algebraic statistics community. One of the motivations for studying toric
ideals of hypergraphs comes from algebraic statistics, where generators of the
toric ideal give a basis for random walks on fibers of the statistical model
specified by the hypergraph. Further, understanding the structure of the
generators gives insight into the model geometry.Comment: Section 3 is new: it explains connections to log-linear models in
algebraic statistics and to combinatorial discrepancy. Section 6 (open
problems) has been moderately revise
Using membrane computing for effective homology
Effective Homology is an algebraic-topological method based on the computational concept of chain homotopy equivalence on a cell complex. Using this algebraic data structure, Effective Homology gives answers to some important computability problems in Algebraic Topology. In a discrete context, Effective Homology can be seen as a combinatorial layer given by a forest graph structure spanning every cell of the complex. In this paper, by taking as input a pixel-based 2D binary object, we present a logarithmic-time uniform solution for describing a chain homotopy operator ϕ for its adjacency graph. This solution is based on Membrane Computing techniques applied to the spanning forest problem and it can be easily extended to higher dimensions
Shadows and intersections: stability and new proofs
We give a short new proof of a version of the Kruskal-Katona theorem due to
Lov\'asz. Our method can be extended to a stability result, describing the
approximate structure of configurations that are close to being extremal, which
answers a question of Mubayi. This in turn leads to another combinatorial proof
of a stability theorem for intersecting families, which was originally obtained
by Friedgut using spectral techniques and then sharpened by Keevash and Mubayi
by means of a purely combinatorial result of Frankl. We also give an algebraic
perspective on these problems, giving yet another proof of intersection
stability that relies on expansion of a certain Cayley graph of the symmetric
group, and an algebraic generalisation of Lov\'asz's theorem that answers a
question of Frankl and Tokushige.Comment: 18 page
Equivalence among optimization problems on matrix sets
AbstractTreatment of optimization problems on matrix sets is a general framework for the study of some large classes of discrete programming problems, for the investigation of connections between different classes of such problems. An appropriate formalism is introduced. It gives a possibility to include in this study bottle-neck problems and other combinatorial optimization problems over totally ordered commutative semigroups. Concepts of equivalency and of weak equivalency are defined and some general equivalency theorems are proved. The main problem under discussion is for which problems an equivalent problem over a finite ordered algebraic structure can be constructed
Identifiability of Points and Rigidity of Hypergraphs under Algebraic Constraints
Identifiability of data is one of the fundamental problems in data science.
Mathematically it is often formulated as the identifiability of points
satisfying a given set of algebraic relations. A key question then is to
identify sufficient conditions for observations to guarantee the
identifiability of the points.
This paper proposes a new general framework for capturing the identifiability
problem when a set of algebraic relations has a combinatorial structure and
develops tools to analyze the impact of the underlying combinatorics on the
local or global identifiability of points. Our framework is built on the
language of graph rigidity, where the measurements are Euclidean distances
between two points, but applicable in the generality of hypergraphs with
arbitrary algebraic measurements. We establish necessary and sufficient
(hyper)graph theoretical conditions for identifiability by exploiting
techniques from graph rigidity theory and algebraic geometry of secant
varieties
Random Sampling in Computational Algebra: Helly Numbers and Violator Spaces
This paper transfers a randomized algorithm, originally used in geometric
optimization, to computational problems in commutative algebra. We show that
Clarkson's sampling algorithm can be applied to two problems in computational
algebra: solving large-scale polynomial systems and finding small generating
sets of graded ideals. The cornerstone of our work is showing that the theory
of violator spaces of G\"artner et al.\ applies to polynomial ideal problems.
To show this, one utilizes a Helly-type result for algebraic varieties. The
resulting algorithms have expected runtime linear in the number of input
polynomials, making the ideas interesting for handling systems with very large
numbers of polynomials, but whose rank in the vector space of polynomials is
small (e.g., when the number of variables and degree is constant).Comment: Minor edits, added two references; results unchange
Computation with Polynomial Equations and Inequalities arising in Combinatorial Optimization
The purpose of this note is to survey a methodology to solve systems of
polynomial equations and inequalities. The techniques we discuss use the
algebra of multivariate polynomials with coefficients over a field to create
large-scale linear algebra or semidefinite programming relaxations of many
kinds of feasibility or optimization questions. We are particularly interested
in problems arising in combinatorial optimization.Comment: 28 pages, survey pape
Combinatorial complexity in o-minimal geometry
In this paper we prove tight bounds on the combinatorial and topological
complexity of sets defined in terms of definable sets belonging to some
fixed definable family of sets in an o-minimal structure. This generalizes the
combinatorial parts of similar bounds known in the case of semi-algebraic and
semi-Pfaffian sets, and as a result vastly increases the applicability of
results on combinatorial and topological complexity of arrangements studied in
discrete and computational geometry. As a sample application, we extend a
Ramsey-type theorem due to Alon et al., originally proved for semi-algebraic
sets of fixed description complexity to this more general setting.Comment: 25 pages. Revised version. To appear in the Proc. London Math. So
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