On the Complexity of an Unregulated Traffic Crossing

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9214)

Abstract

One of the most challenging aspects of traffic coordination involves traffic intersections. In this paper we consider two formulations of a simple and fundamental geometric optimization problem involving coordinating the motion of vehicles through an intersection.

We are given a set of n vehicles in the plane, each modeled as a unit length line segment that moves monotonically, either horizontally or vertically, subject to a maximum speed limit. Each vehicle is described by a start and goal position and a start time and deadline. The question is whether, subject to the speed limit, there exists a collision-free motion plan so that each vehicle travels from its start position to its goal position prior to its deadline.

We present three results. We begin by showing that this problem is \(\mathsf {NP}\)-complete with a reduction from 3-SAT. Second, we consider a constrained version in which cars traveling horizontally can alter their speeds while cars traveling vertically cannot. We present a simple algorithm that solves this problem in \(O(n \log n)\) time. Finally, we provide a solution to the discrete version of the problem and prove its asymptotic optimality in terms of the maximum delay of a vehicle.

Keywords

Speed Limit Maximum Delay Collision Zone Goal Position Lead Vehicle 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Arkin, E.M., Mitchell, J.S.B., Polishchuk, V.: Maximum thick paths in static and dynamic environments. Comput. Geom. Theory Appl. 43(3), 279–294 (2010)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Au, T.-C., Stone, P.: Motion planning algorithms for autonomous intersection management. In: Bridging the Gap Between Task and Motion Planning (2010)Google Scholar
  3. 3.
    Berger, F., Klein, R.: A traveller’s problem. In: Proc. 26th Annu. Sympos. Comput. Geom., SoCG 2010, pp. 176–182. ACM, New York (2010)Google Scholar
  4. 4.
    Carlino, D., Boyles, S.D., Stone, P.: Auction-based autonomous intersection management. In: 2013 16th International IEEE Conference on Intelligent Transportation Systems-(ITSC), pp. 529–534. IEEE (2013)Google Scholar
  5. 5.
    Clarke, G., Wright, J.W.: Scheduling of vehicles from a central depot to a number of delivery points. Operations Res. 12(4), 568–581 (1964)CrossRefGoogle Scholar
  6. 6.
    Dantzig, G.B., Ramser, J.H.: The truck dispatching problem. Management Sci. 6(1), 80–91 (1959)MathSciNetMATHGoogle Scholar
  7. 7.
    Dresner, K., Stone, P.: Multiagent traffic management: a reservation-based intersection control mechanism. In: Proc. Third Internat. Joint Conf. on Auton. Agents and Multi. Agent Syst., pp. 530–537. IEEE Computer Society (2004)Google Scholar
  8. 8.
    Dresner, K., Stone, P.: Multiagent traffic management: an improved intersection control mechanism. In: Proc. Fourth Internat. Joint Conf. on Auton. Agents and Multi. Agent Syst., pp. 471–477. ACM (2005)Google Scholar
  9. 9.
    Dresner, K.M., Stone, P.: A multiagent approach to autonomous intersection management. J. Artif. Intell. Res. 31, 591–656 (2008)Google Scholar
  10. 10.
    Fenton, R.E., Melocik, G.C., Olson, K.W.: On the steering of automated vehicles: Theory and experiment. IEEE Trans. Autom. Control 21(3), 306–315 (1976)CrossRefGoogle Scholar
  11. 11.
    Fiorini, P., Shiller, Z.: Motion planning in dynamic environments using velocity obstacles. Internat. J. Robot. Res. 17(7), 760–772 (1998)CrossRefGoogle Scholar
  12. 12.
    Hearn, R.A., Demaine, E.D.: Pspace-completeness of sliding-block puzzles and other problems through the nondeterministic constraint logic model of computation. Theo. Comp. Sci. 343(1–2), 72–96 (2005)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Petti, S., Fraichard, T.: Safe motion planning in dynamic environments. In: 2005 IEEE/RSJ International Conference on Intelligent Robots and Systems, 2005. (IROS 2005), pp. 2210–2215, August 2005Google Scholar
  14. 14.
    Rajamani, R.: Vehicle Dynamics and Control. Springer Science & Business Media, December 2011Google Scholar
  15. 15.
    Solomon, M.M.: Algorithms for the vehicle routing and scheduling problems with time window constraints. Operations Res. 35(2), 254–265 (1987)CrossRefMATHGoogle Scholar
  16. 16.
    Van Middlesworth, M., Dresner, K., Stone, P.: Replacing the stop sign: unmanaged intersection control for autonomous vehicles. In: Proc. Seventh Internat. Joint Conf. on Auton. Agents and Multi. Agent Syst., pp. 1413–1416. International Foundation for Autonomous Agents and Multiagent Systems (2008)Google Scholar
  17. 17.
    Wurman, P.R., D’Andrea, R., Mountz, M.: Coordinating hundreds of cooperative, autonomous vehicles in warehouses. The AI magazine 29(1), 9–19 (2008)Google Scholar
  18. 18.
    Yu, J., LaValle, S.M.: Multi-agent path planning and network flow. In: Frazzoli, E., Lozano-Perez, T., Roy, N., Rus, D. (eds.) Algorithmic Foundations of Robotics X. Springer Tracts in Advanced Robotics, vol. 86, pp. 157–173. Springer, Heidelberg (2013). http://dx.doi.org/10.1007/978-3-642-36279-8_10

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of MarylandCollege ParkUSA

Personalised recommendations