Harald Räcke

Balanced graph partitioning

In this paper we consider the problem of (k, υ)-balanced graph partitioning - dividing the vertices of a graph into k almost equal size components (each of size less than υ • n k) so that the capacity of edges between different components is minimized. This problem is a natural generalization of several other problems such as minimum bisection, which is the (2,1)-balanced partitioning problem. We present a bicriteria polynomial time approximation algorithm with an O(log2n)-approximation for any constant υ > 1. For υ = 1 we show that no polytime approximation algorithm can guarantee a finite approximation ratio unless P=NP. Previous work has only considered the (k, υ)-balanced partitioning problem for υ ≥ 2.

Minimizing congestion in general networks

A principle task in parallel and distributed systems is to reduce the communication load in the interconnection network, as this is usually the major bottleneck for the performance of distributed applications. We introduce a framework for solving online problems that aim to minimize the congestion (i.e. the maximum load of a network link) in general topology networks. We apply this framework to the problem of online routing of virtual circuits and to a dynamic data management problem. For both scenarios we achieve a competitive ratio of O(log/sup 3/ n) with respect to the congestion of the network links. Our online algorithm for the routing problem has the remarkable property that it is oblivious, i.e., the path chosen for a virtual circuit is independent of the current network load. Oblivious routing strategies can easily be implemented in distributed environments and have therefore been intensively studied for certain network topologies as e.g. meshes, tori and hypercubic networks. This is the first oblivious path selection algorithm that achieves a polylogarithmic competitive ratio in general networks.A principle task in parallel and distributed systems is to reduce the communication load in the interconnection network, as this is usually the major bottleneck for the performance of distributed applications. We introduce a framework for solving online problems that aim to minimize the congestion (i.e. the maximum load of a network link) in general topology networks. We apply this framework to the problem of online routing of virtual circuits and to a dynamic data management problem. For both scenarios we achieve a competitive ratio of O(log/sup 3/ n) with respect to the congestion of the network links. Our online algorithm for the routing problem has the remarkable property that it is oblivious, i.e., the path chosen for a virtual circuit is independent of the current network load. Oblivious routing strategies can easily be implemented in distributed environments and have therefore been intensively studied for certain network topologies as e.g. meshes, tori and hypercubic networks. This is the first oblivious path selection algorithm that achieves a polylogarithmic competitive ratio in general networks.

Optimal hierarchical decompositions for congestion minimization in networks

Hierarchical graph decompositions play an important role in the design of approximation and online algorithms for graph problems. This is mainly due to the fact that the results concerning the approximation of metric spaces by tree metrics (e.g. [10,11,14,16]) depend on hierarchical graph decompositions. In this line of work a probability distribution over tree graphs is constructed from a given input graph, in such a way that the tree distances closely resemble the distances in the original graph. This allows it, to solve many problems with a distance-based cost function on trees, and then transfer the tree solution to general undirected graphs with only a logarithmic loss in the performance guarantee. The results about oblivious routing [30,22] in general undirected graphs are based on hierarchical decompositions of a different type in the sense that they are aiming to approximate the bottlenecks in the network (instead of the point-to-point distances). We call such decompositions cut-based decompositions. It has been shown that they also can be used to design approximation and online algorithms for a wide variety of different problems, but at the current state of the art the performance guarantee goes down by an O(log2n log log n)-factor when making the transition from tree networks to general graphs. In this paper we show how to construct cut-based decompositions that only result in a logarithmic loss in performance, which is asymptotically optimal. Remarkably, one major ingredient of our proof is a distance-based decomposition scheme due to Fakcharoenphol, Rao and Talwar [16]. This shows an interesting relationship between these seemingly different decomposition techniques. The main applications of the new decomposition are an optimal O(log n)-competitive algorithm for oblivious routing in general undirected graphs, and an O(log n)-approximation for Minimum Bisection, which improves the O(log1.5n) approximation by Feige and Krauthgamer [17].

Optimal oblivious routing in polynomial time

A recent seminal result of Racke is that for any undirected network there is an oblivious routing algorithm with a polylogarithmic competitive ratio with respect to congestion. Unfortunately, Rackes construction is not polynomial time. We give a polynomial time construction that guarantees Rackes bounds, and more generally gives the true optimal ratio for any (undirected or directed) network.

Balanced Graph Partitioning

We consider the problem of partitioning a graph into k components of roughly equal size while minimizing the capacity of the edges between different components of the cut. In particular we require that for a parameter ν ≥ 1, no component contains more than ν · n/k of the graph vertices.For k = 2 and ν = 1 this problem is equivalent to the well-known Minimum Bisection problem for which an approximation algorithm with a polylogarithmic approximation guarantee has been presented in [FK]. For arbitrary k and ν ≥ 2 a bicriteria approximation ratio of O(log n) was obtained by Even et al. [ENRS1] using the spreading metrics technique.We present a bicriteria approximation algorithm that for any constant ν > 1 runs in polynomial time and guarantees an approximation ratio of O(log1.5n) (for a precise statement of the main result see Theorem 6). For ν = 1 and k ≥ 3 we show that no polynomial time approximation algorithm can guarantee a finite approximation ratio unless P = NP.

Fast convergence to Wardrop equilibria by adaptive sampling methods

We study rerouting policies in a dynamic round-based variant of a well known game theoretic traffic model due to Wardrop. Previous analyses (mostly in the context of selfish routing) based on Wardrops model focus mostly on the static analysis of equilibria. In this paper, we ask the question whether the population of agents responsible for routing the traffic can jointly compute or better learn a Wardrop equilibrium efficiently. The rerouting policies that we study are of the following kind. In each round, each agent samples an alternative routing path and compares the latency on this path with its current latency. If the agent observes that it can improve its latency then it switches with some probability depending on the possible improvement to the better path.We can show various positive results based on a rerouting policy using an adaptive sampling rule that implicitly amplifies paths that carry a large amount of traffic in the Wardrop equilibrium. For general asymmetric games, we show that a simple replication protocol in which agents adopt strategies of more successful agents reaches a certain kind of bicriteria equilibrium within a time bound that is independent of the size and the structure of the network but only depends on a parameter of the latency functions, that we call the relative slope. For symmetric games, this result has an intuitive interpretation: Replication approximately satisfies almost everyone very quickly.In order to achieve convergence to a Wardrop equilibrium besides replication one also needs an exploration component discovering possibly unused strategies. We present a sampling based replication-exploration protocol and analyze its convergence time for symmetric games. For example, if the latency functions are defined by positive polynomials in coefficient representation, the convergence time is polynomial in the representation length of the latency functions. To the best of our knowledge, all previous results on the speed of convergence towards Wardrop equilibria, even when restricted to linear latency functions, were pseudopolynomial.In addition to the upper bounds on the speed of convergence, we can also present a lower bound demonstrating the necessity of adaptive sampling by showing that static sampling methods result in a slowdown that is exponential in the size of the network. A further lower bound illustrates that the relative slope is, in fact, the relevant parameter that determines the speed of convergence.

Randomized Pursuit-Evasion in Graphs

We analyse a randomized pursuit-evasion game played by two players on a graph, a hunter and a rabbit. Let

A practical algorithm for constructing oblivious routing schemes

G

Approximation Algorithms for Data Management in Networks

be any connected, undirected graph with

Oblivious routing in directed graphs with random demands

n