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New Complexity Results and Algorithms for the Minimum Tollbooth Problem

  • Soumya Basu
  • Thanasis Lianeas
  • Evdokia Nikolova
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9470)

Abstract

The inefficiency of the Wardrop equilibrium of nonatomic routing games can be eliminated by placing tolls on the edges of a network so that the socially optimal flow is induced as an equilibrium flow. A solution where the minimum number of edges are tolled may be preferable over others due to its ease of implementation in real networks. In this paper we consider the minimum tollbooth (\({MINTB}\)) problem, which seeks social optimum inducing tolls with minimum support. We prove for single commodity networks with linear latencies that the problem is NP-hard to approximate within a factor of 1.1377 through a reduction from the minimum vertex cover problem. Insights from network design motivate us to formulate a new variation of the problem where, in addition to placing tolls, it is allowed to remove unused edges by the social optimum. We prove that this new problem remains NP-hard even for single commodity networks with linear latencies, using a reduction from the partition problem. On the positive side, we give the first exact polynomial solution to the \({MINTB}\) problem in an important class of graphs—series-parallel graphs. Our algorithm solves \({MINTB}\) by first tabulating the candidate solutions for subgraphs of the series-parallel network and then combining them optimally.

Notes

Acknowledgement

We would like to thank Steve Boyles and Sanjay Shakkottai for helpful discussions. This work was supported in part by NSF grant numbers CCF-1216103, CCF-1350823 and CCF-1331863.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Open Access This chapter is distributed under the terms of the Creative Commons Attribution Noncommercial License, which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

Authors and Affiliations

  • Soumya Basu
    • 1
  • Thanasis Lianeas
    • 1
  • Evdokia Nikolova
    • 1
  1. 1.University of Texas at AustinAustinUSA

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