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In this thesis, we study multicommodity routing problems in networks, in which commodities have to be routed from source to destination nodes. Such problems model for instance the traffic flows in street networks, data flows in the Internet, or production flows in factories. In most of these applications, the quality of a flow depends on load dependent cost functions on the edges of the given network. The total cost of a flow is usually defined as the sum of the arc cost of the network. An optimal flow minimizes this cost. A main focus of this thesis is to investigate online multicommodity routing problems in networks, in which commodities have to be routed sequentially. Arcs are equipped with load dependent price functions defining routing costs, which have to be minimized. We discuss a greedy online algorithm that routes (fractionally) each commodity by minimizing a convex cost function that depends on the previously routed flow. We present a competitive analysis of this algorithm and prove upper bounds of (d+1)^(d+1) for polynomial price functions with nonnegative coefficients and maximum degree d. For networks with two nodes and parallel arcs, we show that this algorithm returns an optimal solution. Without restrictions on the price functions and network, no algorithm is competitive. We also investigate a variant in which the demands have to be routed unsplittably. In this case, it is NP-hard to compute the offline optimum. Furthermore, we study selfish routing problems (network games). In a network game, players route demand in a network with minimum cost. In this setting, we study the quality of Nash equilibria compared to the the system optimum (price of anarchy) in network games with nonatomic and atomic players and spittable flow. As a main result, we prove upper bounds on the price of anarchy for polynomial latency functions with nonnegative coefficients and maximum degree d, which improve upon the previous best ones.

We consider a multicommodity routing problem, where demands are released \emph{online} and have to be routed in a network during specified time windows. The objective is to minimize a time and load dependent convex cost function of the aggregate arc flow. First, we study the fractional routing variant. We present two online algorithms, called Seq and Seq$^2$. Our first main result states that, for cost functions defined by polynomial price functions with nonnegative coefficients and maximum degree~$d$, the competitive ratio of Seq and Seq$^2$ is at most $(d+1)^{d+1}$, which is tight. We also present lower bounds of $(0.265\,(d+1))^{d+1}$ for any online algorithm. In the case of a network with two nodes and parallel arcs, we prove a lower bound of $(2-\frac{1}{2} \sqrt{3})$ on the competitive ratio for Seq and Seq$^2$, even for affine linear price functions. Furthermore, we study resource augmentation, where the online algorithm has to route less demand than the offline adversary. Second, we consider unsplittable routings. For this setting, we present two online algorithms, called U-Seq and U-Seq$^2$. We prove that for polynomial price functions with nonnegative coefficients and maximum degree~$d$, the competitive ratio of U-Seq and U-Seq$^2$ is bounded by $O{1.77^d\,d^{d+1}}$. We present lower bounds of $(0.5307\,(d+1))^{d+1}$ for any online algorithm and $(d+1)^{d+1}$ for our algorithms. Third, we consider a special case of our framework: online load balancing in the $\ell_p$-norm. For the fractional and unsplittable variant of this problem, we show that our online algorithms are $p$ and $O{p}$ competitive, respectively. Such results where previously known only for scheduling jobs on restricted (un)related parallel machines.

We study online multicommodity minimum cost routing problems in networks, where commodities have to be routed sequentially. Arcs are equipped with load dependent price functions defining the routing weights. We discuss an online algorithm that routes each commodity by minimizing a convex cost function that depends on the demands that are previously routed. We present a competitive analysis of this algorithm showing that for affine linear price functions this algorithm is $4K/2+K$-competitive, where $K$ is the number of commodities. For the parallel arc case this algorithm is optimal. Without restrictions on the price functions and network, no algorithm is competitive. Finally, we investigate a variant in which the demands have to be routed unsplittably.

In this paper we study online multicommodity routing problems in networks, in which commodities have to be routed sequentially. The flow of each commodity can be split on several paths. Arcs are equipped with load dependent price functions defining routing costs, which have to be minimized. We discuss a greedy online algorithm that routes each commodity by minimizing a convex cost function that only depends on the demands previously routed. We present a competitive analysis of this algorithm showing that for affine linear price functions this algorithm is 4K2 (1+K)2 -competitive, where K is the number of commodities. For the single-source single-destination case, this algorithm is optimal. Without restrictions on the price functions and network, no algorithm is competitive. Finally, we investigate a variant in which the demands have to be routed unsplittably.

In this paper, we present a novel approach to the congestion control and resource allocation problem of elastic and real-time traffic in telecommunication networks. With the concept of utility functions, where each source uses a utility function to evaluate the benefit from achieving a transmission rate, we interpret the resource allocation problem as a global optimization problem. The solution to this problem is characterized by a new fairness criterion, \e{utility proportional fairness}. We argue that it is an application level performance measure, i.e. the utility that should be shared fairly among users. As a result of our analysis, we obtain congestion control laws at links and sources that are globally stable and provide a utility proportional fair resource allocation in equilibrium. We show that a utility proportional fair resource allocation also ensures utility max-min fairness for all users sharing a single path in the network. As a special case of our framework, we incorporate utility max-min fairness for the entire network. To implement our approach, neither per-flow state at the routers nor explicit feedback beside ECN (Explicit Congestion Notification) from the routers to the end-systems is required.

In this paper, we study the efficiency of Nash equilibria for a sequence of nonatomic routing games. We assume that the games are played consecutively in time in an online fashion: by the time of playing game $i$, future games $i+1,\dots,n$ are not known, and, once players of game $i$ are in equilibrium, their corresponding strategies and costs remain fixed. Given a sequence of games, the cost for the sequence of Nash equilibria is defined as the sum of the cost of each game. We analyze the efficiency of a sequence of Nash equilibria in terms of competitive analysis arising in the online optimization field. Our main result states that the online algorithm $\sl {SeqNash}$ consisting of the sequence of Nash equilibria is $\frac{4n}{2+n}$-competitive for affine linear latency functions. For $n=1$, this result contains the bound on the price of anarchy of $\frac{4}{3}$ for affine linear latency functions of Roughgarden and Tardos [2002] as a special case. Furthermore, we analyze a problem variant with a modified cost function that reflects the total congestion cost, when all games have been played. In this case, we prove an upper bound of $\frac{4n}{2+n}$ on the competitive ratio of $\sl {SeqNash}$. We further prove a lower bound of $\frac{3n-2}{n}$ of $\sl {SeqNash}$ showing that for $n=2$ our upper bound is tight.