6,265 research outputs found
On the theta number of powers of cycle graphs
We give a closed formula for Lovasz theta number of the powers of cycle
graphs and of their complements, the circular complete graphs. As a
consequence, we establish that the circular chromatic number of a circular
perfect graph is computable in polynomial time. We also derive an asymptotic
estimate for this theta number.Comment: 17 page
Entanglement-assisted zero-error capacity is upper-bounded by the Lovász ϑ function
The zero-error capacity of a classical channel is expressed in terms of the independence number of some graph and its tensor powers. This quantity is hard to compute even for small graphs such as the cycle of length seven, so upper bounds such as the Lovász theta function play an important role in zero-error communication. In this paper, we show that the Lovász theta function is an upper bound on the zero-error capacity even in the presence of entanglement between the sender and receiver
Powers of the theta divisor and relations in the tautological ring
We show that the vanishing of the -st power of the theta divisor on
the universal abelian variety implies, by pulling back along a
collection of Abel--Jacobi maps, the vanishing results in the tautological ring
of of Looijenga, Ionel, Graber--Vakil, and
Faber--Pandharipande. We also show that Pixton's double ramification cycle
relations, which generalize the theta vanishing relations and were recently
proved by the first and third authors, imply Theorem~ of Graber and
Vakil, and we provide an explicit algorithm for expressing any tautological
class on of sufficiently high codimension as a
boundary class.Comment: 29 page
Convex Hulls of Algebraic Sets
This article describes a method to compute successive convex approximations
of the convex hull of a set of points in R^n that are the solutions to a system
of polynomial equations over the reals. The method relies on sums of squares of
polynomials and the dual theory of moment matrices. The main feature of the
technique is that all computations are done modulo the ideal generated by the
polynomials defining the set to the convexified. This work was motivated by
questions raised by Lov\'asz concerning extensions of the theta body of a graph
to arbitrary real algebraic varieties, and hence the relaxations described here
are called theta bodies. The convexification process can be seen as an
incarnation of Lasserre's hierarchy of convex relaxations of a semialgebraic
set in R^n. When the defining ideal is real radical the results become
especially nice. We provide several examples of the method and discuss
convergence issues. Finite convergence, especially after the first step of the
method, can be described explicitly for finite point sets.Comment: This article was written for the "Handbook of Semidefinite, Cone and
Polynomial Optimization: Theory, Algorithms, Software and Applications
Asymptotics of classical spin networks
A spin network is a cubic ribbon graph labeled by representations of
. Spin networks are important in various areas of Mathematics
(3-dimensional Quantum Topology), Physics (Angular Momentum, Classical and
Quantum Gravity) and Chemistry (Atomic Spectroscopy). The evaluation of a spin
network is an integer number. The main results of our paper are: (a) an
existence theorem for the asymptotics of evaluations of arbitrary spin networks
(using the theory of -functions), (b) a rationality property of the
generating series of all evaluations with a fixed underlying graph (using the
combinatorics of the chromatic evaluation of a spin network), (c) rigorous
effective computations of our results for some -symbols using the
Wilf-Zeilberger theory, and (d) a complete analysis of the regular Cube
spin network (including a non-rigorous guess of its Stokes constants), in the
appendix.Comment: 24 pages, 32 figure
Vertex elimination orderings for hereditary graph classes
We provide a general method to prove the existence and compute efficiently
elimination orderings in graphs. Our method relies on several tools that were
known before, but that were not put together so far: the algorithm LexBFS due
to Rose, Tarjan and Lueker, one of its properties discovered by Berry and
Bordat, and a local decomposition property of graphs discovered by Maffray,
Trotignon and Vu\vskovi\'c. We use this method to prove the existence of
elimination orderings in several classes of graphs, and to compute them in
linear time. Some of the classes have already been studied, namely
even-hole-free graphs, square-theta-free Berge graphs, universally signable
graphs and wheel-free graphs. Some other classes are new. It turns out that all
the classes that we study in this paper can be defined by excluding some of the
so-called Truemper configurations. For several classes of graphs, we obtain
directly bounds on the chromatic number, or fast algorithms for the maximum
clique problem or the coloring problem
Recognizing Graph Theoretic Properties with Polynomial Ideals
Many hard combinatorial problems can be modeled by a system of polynomial
equations. N. Alon coined the term polynomial method to describe the use of
nonlinear polynomials when solving combinatorial problems. We continue the
exploration of the polynomial method and show how the algorithmic theory of
polynomial ideals can be used to detect k-colorability, unique Hamiltonicity,
and automorphism rigidity of graphs. Our techniques are diverse and involve
Nullstellensatz certificates, linear algebra over finite fields, Groebner
bases, toric algebra, convex programming, and real algebraic geometry.Comment: 20 pages, 3 figure
Theta Bodies for Polynomial Ideals
Inspired by a question of Lov\'asz, we introduce a hierarchy of nested
semidefinite relaxations of the convex hull of real solutions to an arbitrary
polynomial ideal, called theta bodies of the ideal. For the stable set problem
in a graph, the first theta body in this hierarchy is exactly Lov\'asz's theta
body of the graph. We prove that theta bodies are, up to closure, a version of
Lasserre's relaxations for real solutions to ideals, and that they can be
computed explicitly using combinatorial moment matrices. Theta bodies provide a
new canonical set of semidefinite relaxations for the max cut problem. For
vanishing ideals of finite point sets, we give several equivalent
characterizations of when the first theta body equals the convex hull of the
points. We also determine the structure of the first theta body for all ideals.Comment: 26 pages, 3 figure
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