The multi-commodity flow problem is a classical combinatorial optimization problem that addresses a number of practically important issues of congestion and bandwidth management in connection-oriented network architectures. We consider solutions for distributed multi-commodity flow problems, which are solved by multiple agents operating in a cooperative but uncoordinated manner. We provide the first stateless greedy distributed algorithm for the concurrent multi-commodity flow problem with poly-logarithmic convergence. More precisely, our algorithm achieves 1+ɛ approximation, with running time O(log P·log O(1) m·(1/ɛ) O(1)) where P is the number of flow-paths in the network. No prior results exist for our model. Our algorithm is a reasonable alternative to existing polynomial sequential approximation algorithms, such as Garg-Könemann [17]. The algorithm is simple and can be easily implemented or taught in a classroom. Remarkably, our algorithm requires that the increase in the flow rate on a link is more aggressive than the decrease in the rate. Essentially all of the existing flow-control heuristics are variations of TCP, which uses a conservative cap on the increase (e.g., additive), and a rather liberal cap on the decrease (e.g., multiplicative). In contrast, our algorithm requires the increase to be multiplicative, and that this increase is dramatically more aggressive than the decrease.

the date of receipt and acceptance should be inserted later Abstract We present efficient algorithms for computing approximate Wardrop equilibria in a distributed and concurrent fashion. Our algorithms are exexuted by a finite number of agents each of which controls the flow of one commodity striving to balance the induced latency over all utilised paths. The set of allowed paths is represented by a DAG. Our algorithms are based on previous work on policies for infinite populations of agents. These policies achieve a convergence time which is independent of the underlying network and depends mildly on the latency functions. These policies can neither be applied to a finite set of agents nor can they be simulated directly due to the exponential number of paths. Our algorithms circumvent these problems by computing a randomised path decomposition in every communication round. Based on this decomposition, flow is shifted from overloaded to underloaded paths. This way, our algorithm can handle exponentially large path collections in polynomial time. Our algorithms are stateless, and the number of communication rounds depends polynomially on the approximation quality and is independent of the topology and size of the network.

A canonical distributed optimization problem is solving a Covering/Packing Linear Program in a distributed environment with fast convergence and low communication and space overheads. In this paper, we consider the following covering and packing problems, which are the dual of each other: • Passive Commodity Monitoring: minimize the total cost of monitoring devices used to measure the network traffic on all paths. • Maximum Throughput Multicommodity flow: maximize the total value of the flow with bounded edge capacities. We present the first known distributed algorithms for both of these problems that converge to (1 + ɛ)-approximate solutions in poly-logarithmic time with communication and space overheads that depend on the maximal path length but are almost independent of the size of the entire network. Previous distributed solutions achieving similar approximations required convergence time, communication, or space overheads that depend polynomially on the size of the entire network. The sequential simulation of our algorithm is more efficient than the fastest known approximation algorithms for multicommodity flows, e.g., Garg-Könemann [14], when the maximal path length is small.

We analyze the performance of protocols for load balancing in distributed systems based on no-regret algorithms from online learning theory. These protocols treat load balancing as a repeated game and apply algorithms whose average performance over time is guaranteed to match or exceed the average performance of the best strategy in hindsight. Our approach captures two major aspects of distributed systems. First, in our setting of atomic load balancing, every single process can have a significant impact on the performance and behavior of the system. Furthermore, although in distributed systems participants can query the current state of the system they cannot reliably predict the effect of their actions on it. We address this issue by considering load balancing games in the bulletin board model, where players can find out the delay on all machines, but do not have information on what their experienced delay would have been if they had selected another machine. We show that under these more realistic assumptions, if all players use the wellknown multiplicative weights algorithm, then the quality of the resulting solution is exponentially better than the worst correlated equilibrium, and almost as good as that of the worst Nash. These tighter bounds are derived from analyzing the dynamics of a multi-agent learning system.

Abstract—Advanced channel reservation is emerging as an important feature of ultra high-speed networks requiring the transfer of large files. Applications include scientific data transfers and database backup. In this paper, we present two new, online algorithms for advanced reservation, called BatchAll and BatchLim, that are guaranteed to achieve optimal throughput performance, based on multi-commodity flow arguments. Both algorithms are shown to have polynomial-time complexity and provable bounds on the maximum delay for 1 + ε bandwidth augmented networks. The BatchLim algorithm returns the completion time of a connection immediately as a request is placed, but at the expense of a slightly looser competitive ratio than that of BatchAll. We also present a simple approach that limits the number of parallel paths used by the algorithms while provably bounding the maximum reduction factor in the transmission throughput. We show that, although the number of different paths can be exponentially large, the actual number of paths needed to approximate the flow is quite small and proportional to the number of edges in the network. Simulations for a number of topologies show that, in practice, 3 to 5 parallel paths are sufficient to achieve close to optimal performance. The performance of the competitive algorithms are also compared to a greedy benchmark, both through analysis and simulation. I.

Abstract—Advance channel reservation is emerging as an important feature of ultra high-speed networks requiring the transfer of large files. In this paper, we present two new delaycompetitive algorithms for advance reservation, called BatchAll and BatchLim. These algorithms are guaranteed to achieve optimal throughput performance, based on multi-commodity flow arguments. Unlike BatchAll, the BatchLim algorithm returns the completion time of a connection immediately as a request is placed, but at the expense of a slightly looser competitive ratio than that of BatchAll. We propose a simple approach that limits the number of parallel paths used by the algorithms while provably bounding the maximum reduction factor in the transmission throughput. We show that, although the number of different paths can be exponentially large, the actual number of paths needed to approximate the flow is quite small and proportional to the number of edges in the network. According to our simulations for a number of topologies, three to five parallel paths are sufficient to achieve close to optimal performance. I.

Abstract This work surveys results from [18,19,20,21,22,23]. Recently, the Wardrop model has attracted a lot of attention as a model of selfish behaviour in routing scenarios. In this model, an infinite number of users controls an infinite amount of flow each. The overall flow induces latencies on the edges, and agents strive to minimise their sustained latency selfishly. Most of the studies on this model focus on static properties of equilibria like bounds on the degradation of performance due to the selfishness of the agents and the absence of central coordination, and ways to reduce this degradation. In order to motivate the study of equilibria, one typically uses strong assumptions on the knowledge and rationality of the agents. In this work, we take a different approach by modelling the agents as a dynamic population of agents exchanging their routing path from time to time in order to improve their sustained latency. Our motivation is twofold. First, our results show that it is possible for a population of agents to attain an equilibrium without relying on the abovementioned assumptions, merely by following some simple selfish improvement rules. Second, we use these results to obtain fast distributed algorithms for computing approximate Wardrop equilibria. Our analysis are mainly theoretical, but we also present simulations of dynamic traffic engineering protocols based on our population dynamics. 1

We develop a framework of distributed and stateless solutions for packing and covering linear programs, which are solved by multiple agents operating in a cooperative but uncoordinated manner. Our model has a separate “agent ” controlling each variable and an agent is allowed to read-off the current values only of those constraints in which it has non-zero coefficients. This is a natural model for many distributed applications like flow control, maximum bipartite matching, and dominating sets. The most appealing feature of our algorithms is their simplicity and polylogarithmic convergence. For the packing LP max{c · x | Ax ≤ b, x ≥ 0}, the algorithm associates a dual variable yi = exp [ 1 ɛ ( Aix − 1)] for each constraint i and bi each agent j iteratively increases (resp. decreases) xj multiplicatively

We design completely local, stateless, and self-stabilizing flow control mechanism to be executed by “greedy” agents associated with individual flow paths. Our mechanism is very natural and can be described in a single line: If a path has many “congested ” edges, decrease the flow on the path by a small multiplicative factor, otherwise increase its flow by a small multiplicative factor. The mechanism does not require any initialization or coordination between the agents. We show that starting from an arbitrary feasible flow, the mechanism always maintains feasibility and reaches, after poly-logarithmic number of rounds, a 1 + ɛ approximation of the maximum throughput multicommodity flow. Moreover, the total number of rounds in which the solution is not 1 + ɛ approximate is also poly-logarithmic. Previous distributed solutions in our model either required a state since they used a primal-dual approach or had very slow (polynomial) convergence.