1,137 research outputs found
Computational complexity of reconstruction and isomorphism testing for designs and line graphs
Graphs with high symmetry or regularity are the main source for
experimentally hard instances of the notoriously difficult graph isomorphism
problem. In this paper, we study the computational complexity of isomorphism
testing for line graphs of - designs. For this class of
highly regular graphs, we obtain a worst-case running time of for bounded parameters . In a first step, our approach
makes use of the Babai--Luks algorithm to compute canonical forms of
-designs. In a second step, we show that -designs can be reconstructed
from their line graphs in polynomial-time. The first is algebraic in nature,
the second purely combinatorial. For both, profound structural knowledge in
design theory is required. Our results extend earlier complexity results about
isomorphism testing of graphs generated from Steiner triple systems and block
designs.Comment: 12 pages; to appear in: "Journal of Combinatorial Theory, Series A
Permutation group approach to association schemes
AbstractWe survey the modern theory of schemes (coherent configurations). The main attention is paid to the schurity problem and the separability problem. Several applications of schemes to constructing polynomial-time algorithms, in particular, graph isomorphism tests, are discussed
The complete classification of five-dimensional Dirichlet-Voronoi polyhedra of translational lattices
In this paper we report on the full classification of Dirichlet-Voronoi
polyhedra and Delaunay subdivisions of five-dimensional translational lattices.
We obtain a complete list of affine types (L-types) of Delaunay
subdivisions and it turns out that they are all combinatorially inequivalent,
giving the same number of combinatorial types of Dirichlet-Voronoi polyhedra.
Using a refinement of corresponding secondary cones, we obtain
contraction types. We report on details of our computer assisted enumeration,
which we verified by three independent implementations and a topological mass
formula check.Comment: 16 page
An extensive English language bibliography on graph theory and its applications
Bibliography on graph theory and its application
Neural function approximation on graphs: shape modelling, graph discrimination & compression
Graphs serve as a versatile mathematical abstraction of real-world phenomena in numerous scientific disciplines. This thesis is part of the Geometric Deep Learning subject area, a family of learning paradigms, that capitalise on the increasing volume of non-Euclidean data so as to solve real-world tasks in a data-driven manner. In particular, we focus on the topic of graph function approximation using neural networks, which lies at the heart of many relevant methods. In the first part of the thesis, we contribute to the understanding and design of Graph Neural Networks (GNNs). Initially, we investigate the problem of learning on signals supported on a fixed graph. We show that treating graph signals as general graph spaces is restrictive and conventional GNNs have limited expressivity. Instead, we expose a more enlightening perspective by drawing parallels between graph signals and signals on Euclidean grids, such as images and audio. Accordingly, we propose a permutation-sensitive GNN based on an operator analogous to shifts in grids and instantiate it on 3D meshes for shape modelling (Spiral Convolutions). Following, we focus on learning on general graph spaces and in particular on functions that are invariant to graph isomorphism. We identify a fundamental trade-off between invariance, expressivity and computational complexity, which we address with a symmetry-breaking mechanism based on substructure encodings (Graph Substructure Networks). Substructures are shown to be a powerful tool that provably improves expressivity while controlling computational complexity, and a useful inductive bias in network science and chemistry. In the second part of the thesis, we discuss the problem of graph compression, where we analyse the information-theoretic principles and the connections with graph generative models. We show that another inevitable trade-off surfaces, now between computational complexity and compression quality, due to graph isomorphism. We propose a substructure-based dictionary coder - Partition and Code (PnC) - with theoretical guarantees that can be adapted to different graph distributions by estimating its parameters from observations. Additionally, contrary to the majority of neural compressors, PnC is parameter and sample efficient and is therefore of wide practical relevance. Finally, within this framework, substructures are further illustrated as a decisive archetype for learning problems on graph spaces.Open Acces
Algorithms and the mathematical foundations of computer science
The goal of this chapter is to bring to the attention of philosophers of mathematics the concept of algorithm as it is studied incontemporary theoretical computer science, and at the same time address several foundational questions about the role this notion plays in our practices. A view known as algorithmic realism will be described which maintains that individual algorithms are identical to mathematical objects. Upon considering several ways in which the details of algorithmic realism might be formulated, it will be argued (pace Moschovakis and Gurevich) that there are principled reasons to think that this view cannot be systematically developed in a manner which is compatible with the practice of computational complexity theory and algorithmic analysis
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