How hard is it to find extreme Nash equilibria in network congestion games?

We study the complexity of finding extreme pure Nash equilibria in symmetric (unweighted) network congestion games. In our context best and worst equilibria are those with minimum respectively maximum makespan. On series-parallel graphs a worst Nash equilibrium can be found by a Greedy approach while finding a best equilibrium is NP-hard. For a fixed number of users we give a pseudo-polynomial algorithm to find the best equilibrium in series-parallel networks. For general network topologies also finding a worst equilibrium is NP-hard.

[1]  Gerhard J. Woeginger,et al.  How Hard Is It to Find Extreme Nash Equilibria in Network Congestion Games? , 2008, WINE.

[2]  Christos H. Papadimitriou,et al.  Algorithms, games, and the internet , 2001, STOC '01.

[3]  A. C. Pigou.  Economics of welfare , 1920 .

[4]  Peter Brucker,et al.  Minimum cost flow algorithms for series-parallel networks , 1985, Discret. Appl. Math..

[5]  Paul G. Spirakis,et al.  The structure and complexity of Nash equilibria for a selfish routing game , 2002, Theor. Comput. Sci..

[6]  R. Rosenthal.  A class of games possessing pure-strategy Nash equilibria , 1973 .

[7]  Martin Gairing,et al.  Nashification and the Coordination Ratio for a Selfish Routing Game , 2003, ICALP.

[8]  Christos H. Papadimitriou,et al.  The complexity of pure Nash equilibria , 2004, STOC '04.

[9]  Berthold Vöcking,et al.  Tight bounds for worst-case equilibria , 2002, SODA '02.

[10]  Paul G. Spirakis,et al.  Structure and complexity of extreme Nash equilibria , 2005, Theor. Comput. Sci..

[11]  Paul G. Spirakis,et al.  The price of selfish routing , 2001, STOC '01.

[12]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[13]  Paul G. Spirakis,et al.  Selfish unsplittable flows , 2005, Theor. Comput. Sci..

[14]  Yishay Mansour,et al.  Efficient graph topologies in network routing games , 2009, Games Econ. Behav..

[15]  Elias Koutsoupias,et al.  The price of anarchy of finite congestion games , 2005, STOC '05.

[16]  Paul G. Spirakis,et al.  Symmetry in Network Congestion Games: Pure Equilibria and Anarchy Cost , 2005, WAOA.

[17]  Yossi Azar,et al.  The Price of Routing Unsplittable Flow , 2005, STOC '05.

[18]  Christos H. Papadimitriou,et al.  Worst-case equilibria , 1999 .

[19]  Ravindra K. Ahuja,et al.  Network Flows: Theory, Algorithms, and Applications , 1993 .

[20]  Berthold Vöcking,et al.  On the structure and complexity of worst-case equilibria , 2007, Theor. Comput. Sci..