Oligopolistic Competitive Packet Routing

Oligopolistic competitive packet routing games model situations in which traffic is routed in discrete units through a network over time. We study a game-theoretic variant of packet routing, where in contrast to classical packet routing, we are lacking a central authority to decide on an oblivious routing protocol. Instead, selfish acting decision makers ("players") control a certain amount of traffic each, which needs to be sent as fast as possible from a player-specific origin to a player-specific destination through a commonly used network. The network is represented by a directed graph, each edge of which being endowed with a transit time, as well as a capacity bounding the number of traffic units entering an edge simultaneously. Additionally, a priority policy on the set of players is publicly known with respect to which conflicts at intersections are resolved. We prove the existence of a pure Nash equilibrium and show that it can be constructed by sequentially computing an integral earliest arrival flow for each player. Moreover, we derive several tight bounds on the price of anarchy and the price of stability in single source games

The Complexity of Packing Edge-Disjoint Paths

We introduce and study the complexity of Path Packing. Given a graph G and a list of paths, the task is to embed the paths edge-disjoint in G. This generalizes the well known Hamiltonian-Path problem. Since Hamiltonian Path is efficiently solvable for graphs of small treewidth, we study how this result translates to the much more general Path Packing. On the positive side, we give an FPT-algorithm on trees for the number of paths as parameter. Further, we give an XP-algorithm with the combined parameters maximal degree, number of connected components and number of nodes of degree at least three. Surprisingly the latter is an almost tight result by runtime and parameterization. We show an ETH lower bound almost matching our runtime. Moreover, if two of the three values are constant and one is unbounded the problem becomes NP-hard. Further, we study restrictions to the given list of paths. On the positive side, we present an FPT-algorithm parameterized by the sum of the lengths of the paths. Packing paths of length two is polynomial time solvable, while packing paths of length three is NP-hard. Finally, even the spacial case Exact Path Packing where the paths have to cover every edge in G exactly once is already NP-hard for two paths on 4-regular graphs