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Algorithmica

, Volume 78, Issue 3, pp 788–818 | Cite as

Resolving Braess’s Paradox in Random Networks

  • Dimitris Fotakis
  • Alexis C. KaporisEmail author
  • Thanasis Lianeas
  • Paul G. Spirakis
Article

Abstract

Braess’s paradox states that removing a part of a network may improve the players’ latency at equilibrium. In this work, we study the approximability of the best subnetwork problem for the class of random \({\mathcal {G}}_{n,p}\) instances proven prone to Braess’s paradox by Valiant and Roughgarden RSA ’10 (Random Struct Algorithms 37(4):495–515, 2010), Chung and Young WINE ’10 (LNCS 6484:194–208, 2010) and Chung et al. RSA ’12 (Random Struct Algorithms 41(4):451–468, 2012). Our main contribution is a polynomial-time approximation-preserving reduction of the best subnetwork problem for such instances to the corresponding problem in a simplified network where all neighbors of source s and destination t are directly connected by 0 latency edges. Building on this, we consider two cases, either when the total rate r is sufficiently low, or, when r is sufficiently high. In the first case of low \(r= O(n_{+})\), here \(n_{+}\) is the maximum degree of \(\{s, t\}\), we obtain an approximation scheme that for any constant \(\varepsilon > 0\) and with high probability, computes a subnetwork and an \(\varepsilon \)-Nash flow with maximum latency at most \((1+\varepsilon )L^*+ \varepsilon \), where \(L^*\) is the equilibrium latency of the best subnetwork. Our approximation scheme runs in polynomial time if the random network has average degree \(O(\mathrm {poly}(\ln n))\) and the traffic rate is \(O(\mathrm {poly}(\ln \ln n))\), and in quasipolynomial time for average degrees up to o(n) and traffic rates of \(O(\mathrm {poly}(\ln n))\). Finally, in the second case of high \(r= {\varOmega }(n_{+})\), we compute in strongly polynomial time a subnetwork and an \(\varepsilon \)-Nash flow with maximum latency at most \((1+2\varepsilon + o(1))L^*\).

Keywords

Algorithmic game theory Braess’s paradox Seflish routing Wardrop equilibrium Random graphs 

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Dimitris Fotakis
    • 1
  • Alexis C. Kaporis
    • 2
    Email author
  • Thanasis Lianeas
    • 1
  • Paul G. Spirakis
    • 3
    • 4
  1. 1.Electrical and Computer EngineeringNational Technical University of AthensAthensGreece
  2. 2.Information and Communication Systems DepartmentUniversity of the AegeanSamosGreece
  3. 3.Computer Science DepartmentUniversity of LiverpoolLiverpoolUK
  4. 4.Computer Technology Institute and Press “Diophantus”PatrasGreece

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