3,432 research outputs found
On a packing problem for infinite graphs and independence spaces
In this paper several infinite extensions of the well-known results for packing bases in finite matroids are considered. A counterexample is given to a conjecture of Nash-Williams on edge-disjoint spanning trees of countable graphs, and a sufficient condition is proved for the packing problem in independence spaces over a countably infinite set. © 1979
A semidefinite programming hierarchy for packing problems in discrete geometry
Packing problems in discrete geometry can be modeled as finding independent
sets in infinite graphs where one is interested in independent sets which are
as large as possible. For finite graphs one popular way to compute upper bounds
for the maximal size of an independent set is to use Lasserre's semidefinite
programming hierarchy. We generalize this approach to infinite graphs. For this
we introduce topological packing graphs as an abstraction for infinite graphs
coming from packing problems in discrete geometry. We show that our hierarchy
converges to the independence number.Comment: (v2) 25 pages, revision based on suggestions by referee, accepted in
Mathematical Programming Series B special issue on polynomial optimizatio
Mathematical optimization for packing problems
During the last few years several new results on packing problems were
obtained using a blend of tools from semidefinite optimization, polynomial
optimization, and harmonic analysis. We survey some of these results and the
techniques involved, concentrating on geometric packing problems such as the
sphere-packing problem or the problem of packing regular tetrahedra in R^3.Comment: 17 pages, written for the SIAG/OPT Views-and-News, (v2) some updates
and correction
Axioms for infinite matroids
We give axiomatic foundations for non-finitary infinite matroids with
duality, in terms of independent sets, bases, circuits, closure and rank. This
completes the solution to a problem of Rado of 1966.Comment: 33 pp., 2 fig
A Combinatorial Approach to Nonlocality and Contextuality
So far, most of the literature on (quantum) contextuality and the
Kochen-Specker theorem seems either to concern particular examples of
contextuality, or be considered as quantum logic. Here, we develop a general
formalism for contextuality scenarios based on the combinatorics of hypergraphs
which significantly refines a similar recent approach by Cabello, Severini and
Winter (CSW). In contrast to CSW, we explicitly include the normalization of
probabilities, which gives us a much finer control over the various sets of
probabilistic models like classical, quantum and generalized probabilistic. In
particular, our framework specializes to (quantum) nonlocality in the case of
Bell scenarios, which arise very naturally from a certain product of
contextuality scenarios due to Foulis and Randall. In the spirit of CSW, we
find close relationships to several graph invariants. The recently proposed
Local Orthogonality principle turns out to be a special case of a general
principle for contextuality scenarios related to the Shannon capacity of
graphs. Our results imply that it is strictly dominated by a low level of the
Navascu\'es-Pironio-Ac\'in hierarchy of semidefinite programs, which we also
apply to contextuality scenarios.
We derive a wealth of results in our framework, many of these relating to
quantum and supraquantum contextuality and nonlocality, and state numerous open
problems. For example, we show that the set of quantum models on a
contextuality scenario can in general not be characterized in terms of a graph
invariant.
In terms of graph theory, our main result is this: there exist two graphs
and with the properties \begin{align*} \alpha(G_1) &= \Theta(G_1),
& \alpha(G_2) &= \vartheta(G_2), \\[6pt] \Theta(G_1\boxtimes G_2) & >
\Theta(G_1)\cdot \Theta(G_2),& \Theta(G_1 + G_2) & > \Theta(G_1) + \Theta(G_2).
\end{align*}Comment: minor revision, same results as in v2, to appear in Comm. Math. Phy
On the number of types in sparse graphs
We prove that for every class of graphs which is nowhere dense,
as defined by Nesetril and Ossona de Mendez, and for every first order formula
, whenever one draws a graph and a
subset of its nodes , the number of subsets of which are of
the form
for some valuation of in is bounded by
, for every . This provides
optimal bounds on the VC-density of first-order definable set systems in
nowhere dense graph classes.
We also give two new proofs of upper bounds on quantities in nowhere dense
classes which are relevant for their logical treatment. Firstly, we provide a
new proof of the fact that nowhere dense classes are uniformly quasi-wide,
implying explicit, polynomial upper bounds on the functions relating the two
notions. Secondly, we give a new combinatorial proof of the result of Adler and
Adler stating that every nowhere dense class of graphs is stable. In contrast
to the previous proofs of the above results, our proofs are completely
finitistic and constructive, and yield explicit and computable upper bounds on
quantities related to uniform quasi-wideness (margins) and stability (ladder
indices)
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