134 research outputs found
A coding problem for pairs of subsets
Let be an --element finite set, an integer. Suppose that
and are pairs of disjoint -element subsets of
(that is, , , ). Define the distance of these pairs by . This is the
minimum number of elements of one has to move to obtain the other
pair . Let be the maximum size of a family of pairs of
disjoint subsets, such that the distance of any two pairs is at least .
Here we establish a conjecture of Brightwell and Katona concerning an
asymptotic formula for for are fixed and . Also,
we find the exact value of in an infinite number of cases, by using
special difference sets of integers. Finally, the questions discussed above are
put into a more general context and a number of coding theory type problems are
proposed.Comment: 11 pages (minor changes, and new citations added
Matchings, hypergraphs, association schemes, and semidefinite optimization
We utilize association schemes to analyze the quality of semidefinite
programming (SDP) based convex relaxations of integral packing and covering
polyhedra determined by matchings in hypergraphs. As a by-product of our
approach, we obtain bounds on the clique and stability numbers of some regular
graphs reminiscent of classical bounds by Delsarte and Hoffman. We determine
exactly or provide bounds on the performance of Lov\'{a}sz-Schrijver SDP
hierarchy, and illustrate the usefulness of obtaining commutative subschemes
from non-commutative schemes via contraction in this context
Deep Graph Matching via Blackbox Differentiation of Combinatorial Solvers
Building on recent progress at the intersection of combinatorial optimization
and deep learning, we propose an end-to-end trainable architecture for deep
graph matching that contains unmodified combinatorial solvers. Using the
presence of heavily optimized combinatorial solvers together with some
improvements in architecture design, we advance state-of-the-art on deep graph
matching benchmarks for keypoint correspondence. In addition, we highlight the
conceptual advantages of incorporating solvers into deep learning
architectures, such as the possibility of post-processing with a strong
multi-graph matching solver or the indifference to changes in the training
setting. Finally, we propose two new challenging experimental setups. The code
is available at https://github.com/martius-lab/blackbox-deep-graph-matchingComment: ECCV 2020 conference pape
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Combinatorics
This is the report on the Oberwolfach workshop on Combinatorics, held 1–7 January 2006. Combinatorics is a branch of mathematics studying families of mainly, but not exclusively, finite or countable structures – discrete objects. The discrete objects considered in the workshop were graphs, set systems, discrete geometries, and matrices. The programme consisted of 15 invited lectures, 18 contributed talks, and a problem session focusing on recent developments in graph theory, coding theory, discrete geometry, extremal combinatorics, Ramsey theory, theoretical computer science, and probabilistic combinatorics
The Sketching Complexity of Graph and Hypergraph Counting
Subgraph counting is a fundamental primitive in graph processing, with
applications in social network analysis (e.g., estimating the clustering
coefficient of a graph), database processing and other areas. The space
complexity of subgraph counting has been studied extensively in the literature,
but many natural settings are still not well understood. In this paper we
revisit the subgraph (and hypergraph) counting problem in the sketching model,
where the algorithm's state as it processes a stream of updates to the graph is
a linear function of the stream. This model has recently received a lot of
attention in the literature, and has become a standard model for solving
dynamic graph streaming problems.
In this paper we give a tight bound on the sketching complexity of counting
the number of occurrences of a small subgraph in a bounded degree graph
presented as a stream of edge updates. Specifically, we show that the space
complexity of the problem is governed by the fractional vertex cover number of
the graph . Our subgraph counting algorithm implements a natural vertex
sampling approach, with sampling probabilities governed by the vertex cover of
. Our main technical contribution lies in a new set of Fourier analytic
tools that we develop to analyze multiplayer communication protocols in the
simultaneous communication model, allowing us to prove a tight lower bound. We
believe that our techniques are likely to find applications in other settings.
Besides giving tight bounds for all graphs , both our algorithm and lower
bounds extend to the hypergraph setting, albeit with some loss in space
complexity
Metric Dimension of a Diagonal Family of Generalized Hamming Graphs
Classical Hamming graphs are Cartesian products of complete graphs, and two
vertices are adjacent if they differ in exactly one coordinate. Motivated by
connections to unitary Cayley graphs, we consider a generalization where two
vertices are adjacent if they have no coordinate in common. Metric dimension of
classical Hamming graphs is known asymptotically, but, even in the case of
hypercubes, few exact values have been found. In contrast, we determine the
metric dimension for the entire diagonal family of -dimensional generalized
Hamming graphs. Our approach is constructive and made possible by first
characterizing resolving sets in terms of forbidden subgraphs of an auxiliary
edge-colored hypergraph.Comment: 19 pages, 7 figure
Asymmetric Lee Distance Codes for DNA-Based Storage
We consider a new family of codes, termed asymmetric Lee distance codes, that
arise in the design and implementation of DNA-based storage systems and systems
with parallel string transmission protocols. The codewords are defined over a
quaternary alphabet, although the results carry over to other alphabet sizes;
furthermore, symbol confusability is dictated by their underlying binary
representation. Our contributions are two-fold. First, we demonstrate that the
new distance represents a linear combination of the Lee and Hamming distance
and derive upper bounds on the size of the codes under this metric based on
linear programming techniques. Second, we propose a number of code
constructions which imply lower bounds
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