Abstract
ABSTRACT This paper provides an overview of the state-of-the art and the current research trends concerning shortest paths problem on dynamic graphs. The discussion is divided in two main topics: reoptimization and time-dependent shortest paths. Reoptimization consists in the solution of a sequence of shortest path problems in which each instance slightly differs from the previous one. The reoptimization tackles this problem wisely using information stored in an optimal solution previously computed. On the other hand, shortest path problems on time-dependent graphs are characterized by a weight function which not only depends upon the arcs but changes in time according to a certain time horizon.
Highlights
One of the most iconic algorithms in combinatorial optimization is due to Dijkstra [21], who in 1959 devised a label setting algorithm for the shortest path problem (SPP)
Reoptimizing shortest paths on dynamic graphs consists in solving a sequence of shortest path problems, where each problem differs only slightly from the previous one, because the origin node has been changed, some arcs have been removed from the graph, or the cost of a subset of arcs has been modified
Some arcs weight have been increased or decreased. This problem can be addressed as a shortest path reoptimization problem [26], which consists in solving a sequence of shortest path problems, where the kth problem marginally differs from the (k − 1)th one
Summary
One of the most iconic algorithms in combinatorial optimization is due to Dijkstra [21], who in 1959 devised a label setting algorithm for the shortest path problem (SPP). A clever way to approach it is to design ad hoc algorithms that efficiently use information resulting from previous computations Another type of dynamic graph is the time-dependent graph, introduced in Cooke & Halsey [13], where the characteristic cost function w is defined for each edge (i, j ), as wi j (t ), where t is a time variable in a time domain T. Given a node r ∈ V , named root, the goal of the problem is to find a shortest path from r to all other nodes i ∈ V , i = r. Defining m Boolean decision variables xi j , ∀ (i, j ) ∈ A, the mathematical formulation of the (SPT) problem is the following:.
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