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On the Complexity of Time-Dependent Shortest Paths

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Abstract

We investigate the complexity of shortest paths in time-dependent graphs where the costs of edges (that is, edge travel times) vary as a function of time, and as a result the shortest path between two nodes s and d can change over time. Our main result is that when the edge cost functions are (polynomial-size) piecewise linear, the shortest path from s to d can change n Θ(logn) times, settling a several-year-old conjecture of Dean (Technical Reports, 1999, 2004). However, despite the fact that the arrival time function may have superpolynomial complexity, we show that a minimum delay path for any departure time interval can be computed in polynomial time. We also show that the complexity is polynomial if the slopes of the linear function come from a restricted class and describe an efficient scheme for computing a (1+ϵ)-approximation of the travel time function.

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Notes

  1. One can show that if the true travel time functions for edges are linear but are approximated discretely using fixed-size buckets, then the error in the travel time estimate can grow exponentially, depending on the slope of the linear functions.

  2. Discontinuities in A[e] map to horizontal edges in A[e r] and vice versa. For example, if an edge e has a discontinuous function A[e], i.e., A[e](τ −)=x and A[e](τ +)=y with x<y, then in A[e r] each time in the range [x,y] maps to τ: when traveling from s, one must arrive at edge e at or before t=τ in order to reach the other end of e at or before any time in the range [x,y].

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Authors and Affiliations

  1. Department of Computer Science, University of California, Santa Barbara, CA, 93106, USA

    Luca Foschini & Subhash Suri

  2. Mentor Graphics Corp, 8005 SW Boeckman Rd., Wilsonville, OR, 97070, USA

    John Hershberger

Authors
  1. Luca Foschini
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  2. John Hershberger
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  3. Subhash Suri
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Correspondence to Subhash Suri.

Additional information

A preliminary version of this paper was presented at the 22nd ACM-SIAM Symposium on Discrete Algorithms (SODA), 2011.

This work was partially supported by National Science Foundation grants CNS-1035917 and CCF-0514738.

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Foschini, L., Hershberger, J. & Suri, S. On the Complexity of Time-Dependent Shortest Paths. Algorithmica 68, 1075–1097 (2014). https://doi.org/10.1007/s00453-012-9714-7

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