This paper considers the problem of designing fast, approximate, combinatorial algorithms for multicommodity flows and other fractional packing problems. We provide a different approach to these problems which yields faster and much simpler algorithms. Our approach also allows us to substitute shortest path computations for min-cost flow computations in computing maximum concurrent flow and min-cost multicommodity flow; this yields much faster algorithms when the number of commodities is large.

We develop a framework that allows us to address the issues of admission control and routing in high-speed networks under the restriction that once a call is admitted and routed, it has to proceed to completion and no reroutings are allowed. The "no rerouting" restriction appears in all the proposals for future high-speed networks and stems from current hardware limitations, in particular the fact that the bandwidth-delay product of the newly developed optical communication links far exceeds the buffer capacity of the network. In case the goal is to maximize the throughput, our framework yields an on-line O(lognT )- competitive strategy, where n is the number of nodes in the network and T is the maximum call duration. In other words, our strategy results in throughput that is within O(log nT ) factor of the highest possible throughput achievable by an omniscient algorithm that knows all of the requests in advance. Moreover, we show that no on-line strategy can achieve a better competit...

We give a O( log n)-approximation algorithm for sparsest cut, balanced separator, and graph conductance problems. This improves the O(log n)-approximation of Leighton and Rao (1988). We use a well-known semidefinite relaxation with triangle inequality constraints. Central to our analysis is a geometric theorem about projections of point sets in , whose proof makes essential use of a phenomenon called measure concentration.

The construction of disjoint paths in a network is a basic issue in combinatorial optimization: given a network, and specified pairs of nodes in it, we are interested in finding disjoint paths between as many of these pairs as possible. This leads to a variety of classical NP-complete problems for which very little is known from the point of view of approximation algorithms. It has recently been brought into focus in work on problems such as VLSI layout and routing in high-speed networks; in these settings, the current lack of understanding of the disjoint paths problem is often an obstacle to the design of practical heuristics.

We study the close connections between game theory, on-line prediction and boosting. After a brief review of game theory, we describe an algorithm for learning to play repeated games based on the on-line prediction methods of Littlestone and Warmuth. The analysis of this algorithm yields a simple proof of von Neumann's famous minmax theorem, as well as a provable method of approximately solving a game. We then show that the on-line prediction model is obtained by applying this gameplaying algorithm to an appropriate choice of game and that boosting is obtained by applying the same algorithm to the "dual" of this game. 1 INTRODUCTION The purpose of this paper is to bring out the close connections between game theory, on-line prediction and boosting. Briefly, game theory is the study of games and other interactions of various sorts. On-line prediction is a learning model in which an agent predicts the classification of a sequence of items and attempts to minimize the total number of pre...

Markov decision problems (MDPs) provide the foundations for a number of problems of interest to AI researchers studying automated planning and reinforcement learning. In this paper, we summarize results regarding the complexity of solving MDPs and the running time of MDP solution algorithms. We argue that, although MDPs can be solved efficiently in theory, more study is needed to reveal practical algorithms for solving large problems quickly. To encourage future research, we sketch some alternative methods of analysis that rely on the structure of MDPs.

Communication in all-optical networks requires novel routing paradigms. The high bandwidth of the optic fiber is utilized through wavelengthdivision multiplexing: a single physical optical link can carry several logical signals, provided that they are transmitted on different wavelengths. We study the problem of routing a set of requests (each of which is a pair of nodes to be connected by a path) on sparse networks using a limited number of wavelengths, ensuring that different paths using the same wavelength never use the same physical link. The constraints on the selection of paths and wavelengths depend on the type of photonic switches used in the network. We present eflicient routing techniques for the two types of photonic switches that dominate current research in all-optical networks. Our results es-

. A central drawback of primal-dual interior point methods for semidefinite programs is their lack of ability to exploit problem structure in cost and coefficient matrices. This restricts applicability to problems of small dimension. Typically semidefinite relaxations arising in combinatorial applications have sparse and well structured cost and coefficient matrices of huge order. We present a method that allows to compute acceptable approximations to the optimal solution of large problems within reasonable time. Semidefinite programming problems with constant trace on the primal feasible set are equivalent to eigenvalue optimization problems. These are convex nonsmooth programming problems and can be solved by bundle methods. We propose replacing the traditional polyhedral cutting plane model constructed from subgradient information by a semidefinite model that is tailored for eigenvalue problems. Convergence follows from the traditional approach but a proof is included for completene...

this paper, we present a simple algorithm for solving this problem, and give a simple analysis of the algorithm. The bounds we obtain are not asymptotic and hold for any finite number of rounds. The algorithm and its analysis are based directly on the "on-line prediction" methods of Littlestone and Warmuth [24]. The analysis of this algorithm yields a new (as far as we know) and simple proof of von Neumann's minmax theorem, as well as a provable method of approximately solving a game. We also give more refined variants of the algorithm for this purpose, and we show that one of these is optimal in a very strong sense. The paper is organized as follows. In Section 2 we define the mathematical setup and notation. In Section 3 we introduce the basic multiplicative weights algorithm whose average performance is guaranteed to be almost as good as that of the best fixed mixed strategy. In Section 4 we outline the relationship between our work and some of the extensive existing work on the use of multiplicative weights algorithms for on-line prediction. In Section 5 we show how the algorithm can be used to give a simple proof of Von-Neumann's min-max theorem. In Section 6 we give a version of the algorithm whose distributions are guaranteed to converge to an optimal mixed strategy. We note the possible application of this algorithm to solving linear programming problems and reference other work that have used multiplicative weights to this end. Finally, in Section 7 we show that the convergence rate of the second version of the algorithm is asymptotically optimal. 2 Playing repeated games

We describe fully polynomial time approximation schemes for various multicommodity flow problems in graphs with m edges and n vertices. We present the first approximation scheme for maximum multicommodity flow that is independent of the number of commodities k, and our algorithm improves upon the runtime of previous algorithms by this factor of k, performing in O (ffl \Gamma2 m 2 ) time. For maximum concurrent flow, and minimum cost concurrent flow, we present algorithms that are faster than the current known algorithms when the graph is sparse or the number of commodities k is large, i.e. k ? m=n. Our algorithms build on the framework proposed by Garg and Konemann [4]. They are simple, deterministic, and for the versions without costs, they are strongly polynomial. Our maximum multicommodity flow algorithm extends to an approximation scheme for the maximum weighted multicommodity flow, which is faster than those implied by previous algorithms by a factor of k= log W where W is ...