TY - CHAP

T1 - Computational complexity, fairness, and the price of anarchy of the maximum latency problem extended abstract

AU - Correa, José R.

AU - Schulz, Andreas S.

AU - Stier Moses, Nicolás E.

PY - 2004

Y1 - 2004

N2 - We study the problem of minimizing the maximum latency of flows in networks with congestion. We show that this problem is NP-hard, even when all arc latency functions are linear and there is a single source and sink. Still, one can prove that an optimal flow and an equilibrium flow share a desirable property in this situation: all flow-carrying paths have the same length; i.e., these solutions are "fair," which is in general not true for the optimal flow in networks with nonlinear latency functions. In addition, the maximum latency of the Nash equilibrium, which can be computed efficiently, is within a constant factor of that of an optimal solution. That is, the so-called price of anarchy is bounded. In contrast, we present a family of instances that shows that the price of anarchy is unbounded for instances with multiple sources and a single sink, even in networks with linear latencies. Finally, we show that an s-t-flow that is optimal with respect to the average latency objective is near optimal for the maximum latency objective, and it is close to being fair. Conversely, the average latency of a flow minimizing the maximum latency is also within a constant factor of that of a flow minimizing the average latency.

AB - We study the problem of minimizing the maximum latency of flows in networks with congestion. We show that this problem is NP-hard, even when all arc latency functions are linear and there is a single source and sink. Still, one can prove that an optimal flow and an equilibrium flow share a desirable property in this situation: all flow-carrying paths have the same length; i.e., these solutions are "fair," which is in general not true for the optimal flow in networks with nonlinear latency functions. In addition, the maximum latency of the Nash equilibrium, which can be computed efficiently, is within a constant factor of that of an optimal solution. That is, the so-called price of anarchy is bounded. In contrast, we present a family of instances that shows that the price of anarchy is unbounded for instances with multiple sources and a single sink, even in networks with linear latencies. Finally, we show that an s-t-flow that is optimal with respect to the average latency objective is near optimal for the maximum latency objective, and it is close to being fair. Conversely, the average latency of a flow minimizing the maximum latency is also within a constant factor of that of a flow minimizing the average latency.

UR - http://www.scopus.com/inward/record.url?scp=33745585248&partnerID=8YFLogxK

U2 - 10.1007/978-3-540-25960-2_5

DO - 10.1007/978-3-540-25960-2_5

M3 - Chapter

AN - SCOPUS:33745585248

SN - 3540221131

SN - 9783540221135

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 59

EP - 73

BT - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

A2 - Bienstock, Daniel

A2 - Nemhauser, George

PB - Springer Verlag

ER -