3,968 research outputs found
Developments in the theory of randomized shortest paths with a comparison of graph node distances
There have lately been several suggestions for parametrized distances on a
graph that generalize the shortest path distance and the commute time or
resistance distance. The need for developing such distances has risen from the
observation that the above-mentioned common distances in many situations fail
to take into account the global structure of the graph. In this article, we
develop the theory of one family of graph node distances, known as the
randomized shortest path dissimilarity, which has its foundation in statistical
physics. We show that the randomized shortest path dissimilarity can be easily
computed in closed form for all pairs of nodes of a graph. Moreover, we come up
with a new definition of a distance measure that we call the free energy
distance. The free energy distance can be seen as an upgrade of the randomized
shortest path dissimilarity as it defines a metric, in addition to which it
satisfies the graph-geodetic property. The derivation and computation of the
free energy distance are also straightforward. We then make a comparison
between a set of generalized distances that interpolate between the shortest
path distance and the commute time, or resistance distance. This comparison
focuses on the applicability of the distances in graph node clustering and
classification. The comparison, in general, shows that the parametrized
distances perform well in the tasks. In particular, we see that the results
obtained with the free energy distance are among the best in all the
experiments.Comment: 30 pages, 4 figures, 3 table
New Routing Problems with possibly correlated travel times
In the literature of operational research, Vehicle Routing Problems (VRP) were and still are subject of countless studies.
Under the scope of combinatorial optimization, this thesis analyses some variants of VRP both with deterministic and uncertain travel times.
The deterministic problem under study is a drayage problem with characteristics con- cerning service types and requirement seldom investigated all together. The formulations proposed to model this problem are: the node-arc formulation and the Set Partitioning formu- lation. Concerning the solution methods, two heuristics and a branch-and-price approach are presented.
The section dealing with uncertain and correlated travel times faces two classes of VRP with time windows using either single or joint chance constraints depending on whether missing a customers time window makes the entire route infeasible or not.
From a comparison between deterministic and stochastic methods, these last represent a profitable investment to guarantee the feasibility of the solution in realistic instances
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