575,851 research outputs found
Learning flexible representations of stochastic processes on graphs
Graph convolutional networks adapt the architecture of convolutional neural
networks to learn rich representations of data supported on arbitrary graphs by
replacing the convolution operations of convolutional neural networks with
graph-dependent linear operations. However, these graph-dependent linear
operations are developed for scalar functions supported on undirected graphs.
We propose a class of linear operations for stochastic (time-varying) processes
on directed (or undirected) graphs to be used in graph convolutional networks.
We propose a parameterization of such linear operations using functional
calculus to achieve arbitrarily low learning complexity. The proposed approach
is shown to model richer behaviors and display greater flexibility in learning
representations than product graph methods
On path ranking in time-dependent graphs
In this paper we study a property of time-dependent graphs, dubbed path
ranking invariance. Broadly speaking, a time-dependent graph is path ranking
invariant if the ordering of its paths (w.r.t. travel time) is independent of
the start time. In this paper we show that, if a graph is path ranking
invariant, the solution of a large class of time-dependent vehicle routing
problems can be obtained by solving suitably defined (and simpler)
time-independent routing problems. We also show how this property can be
checked by solving a linear program. If the check fails, the solution of the
linear program can be used to determine a tight lower bound. In order to assess
the value of these insights, the lower bounds have been embedded into an
enumerative scheme. Computational results on the time-dependent versions of the
\textit{Travelling Salesman Problem} and the \textit{Rural Postman Problem}
show that the new findings allow to outperform state-of-the-art algorithms.Comment: 28 pages, 2 figure
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