Brian C. Dean Assistant Professor Division of Computer Science School of Computing, Clemson University Office: McAdams Hall, Room 205 Phone: 864-656-5866 Mail: Brian C. Dean, School of Computing, Box 340974, Clemson SC 29634-0974 Email: bcdean@cs.clemson.edu

Continuous-Time Dynamic Shortest Path Algorithms

We consider the problem of computing shortest paths through a dynamic network -- a network with time-varying characteristics, such as arc travel times and costs, which are known for all values of time. Many types of networks, most notably transportation networks, exhibit such predictable dynamic behavior over the course of time. Dynamic shortest path problems are currently solved in practice by algorithms which operate within a discrete-time framework. In this thesis, we introduce a new set of algorithms for computing shortest paths in continuous-time dynamic networks, and demonstrate for the first time in the literature the feasibility and the advantages of solving dynamic shortest path problems in continuous time. We assume that all time-dependent network data functions are given as piece-wise linear functions of time, a representation capable of easily modeling most common dynamic problems. Additionally, this form of representation and the solution algorithms developed in this thesis are well suited for many augmented static problems such as time-constrained minimum-cost shortest path problems and shortest path problems with time windows.

We discuss the classification, formulation, and mathematical properties of all common variants of the continuous-time dynamic shortest path problem. Two classes of solution algorithms are introduced, both of which are shown to solve all variants of the problem. In problems where arc travel time functions exhibit First-In-First-Out (FIFO) behavior, we show that these algorithms have polynomial running time (see erratum below); although the general problem is NP-hard, we argue that the average-case running time for many common problems should be quite reasonable. Computational results are given which support the theoretical analysis of these algorithms, and which provide a comparison with existing discrete-time algorithms; in most cases, continuous-time approaches are shown to be much more efficient, both in running time and storage requirements, than their discretetime counterparts. Finally, in order to further reduce computation time, we introduce parallel algorithms, and hybrid continuous-discrete approximation algorithms which exploit favorable characteristics of algorithms from both domains.

Erratum: The proofs of Lemmas 4.9 and 4.10 are not correct, and it is still not known whether the end result of these Lemmas is true (i.e., whether our output will have polynomial size). Super-polynomial worst-case lower bounds for the related parametric shortest path problem may suggest that similar bounds may also hold for time-dependent shortest path problems. See the short survey paper "Shortest Paths in FIFO Time-Dependent Networks: Theory and Algorithms" for more information -- this paper presents a more concise exposition of most of the interesting aspects of my thesis.

Available in pdf and postscript.