Abstract. A Nash equilibrium of a routing game is a stable state where no (randomizing) user could benefit from a unilateral deviation. We consider the simplest case of the parallel links network, where links are related. The Social Cost of a Nash equilibrium is the expected maximum latency. We seek the worst-case Nash equilibrium [E. Koutsoupias and C. H. Papadimitriou, Comput. Sci. Rev., 3 (2009), pp. 65–69], which maximizes Social Cost. We continue the study of the fully mixed Nash equilibrium conjecture, abbreviated as the FMNE Conjecture, stating that the worst-case Nash equilibrium is the fully mixed Nash equilibrium, where each user assigns strictly positive probability to every link. Through an extensive combinatorial analysis, we confirm the FMNE Conjecture for the two basic cases where there are either (i) two users on related links, or (ii) many users on two identical links.

We consider the problem of routing n users on m parallel links, under the restriction that each user may only be routed on a link from a certain set of allowed links for the user. Thus, the problem is equivalent to the correspondingly restricted problem of assigning n jobs to m parallel machines. In a pure Nash equilibrium, no user may improve its own individual cost (delay) by unilaterally switching to another link from its set of allowed links. As our main result, we introduce a polynomial time algorithm to compute from any given assignment a pure Nash equilibrium with non-increased makespan. The algorithm gradually changes a given assignment by pushing unsplittable user traffics through a network that is defined by the users and the links. Here, we use ideas from blocking flows. Furthermore, we use similar techniques as in the generic Preflow-Push algorithm to approximate a schedule with minimum makespan, gaining an improved approximation factor of 2 - 1/w1 for identical links, where w1 is the largest user traffic. We extend this result to related links, gaining an approximation factor of 2. Our approximation algorithms run in polynomial time. We close with tight upper bounds on the coordination ratio for pure Nash equilibria.

When executing processes on parallel computer systems they encounter as a major bottleneck inter-processor communication. One way to address this problem is to minimize the communication between processes that are mapped to different processors. This translates to the k-partitioning problem of the corresponding process graph, where k is the number of processors. The classical spectral lower bound of ¦V¦ ÷ 2k Σ k i =1 λi for the k-section width of a graph is well-known. We show new relations between the structure and the eigen values of a graph and present a new method to get tighter lower bounds on the k-section width. This method makes use of the level structure defined by the k-section. We define some global expansion property and prove that for graphs with the same k-section width the spectral lower bound increases with this global expansion. We also present examples of graphs for which our new bounds are tight up to a constant factor.