Lisa Fleischer

A combinatorial strongly polynomial algorithm for minimizing submodular functions

This paper presents a combinatorial polynomial-time algorithm for minimizing submodular functions, answering an open question posed in 1981 by Grötschel, Lovász, and Schrijver. The algorithm employs a scaling scheme that uses a flow in the complete directed graph on the underlying set with each arc capacity equal to the scaled parameter. The resulting algorithm runs in time bounded by a polynomial in the size of the underlying set and the length of the largest absolute function value. The paper also presents a strongly polynomial version in which the number of steps is bounded by a polynomial in the size of the underlying set, independent of the function values.

Approximating Fractional Multicommodity Flow Independent of the Number of Commodities

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 run time of previous algorithms by this factor of k, running in

Tight approximation algorithms for maximum general assignment problems

{{\cal O}^*(\epsilon^{-2}m^2)}

Tolls for heterogeneous selfish users in multicommodity networks and generalized congestion games

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 [Proceedings of the 39th Annual IEEE Symposium on Foundations of Computer Science, IEEE, New York, 1998, pp. 300--309]. They are simple, deterministic, and for the versions without costs, they are strongly polynomial. The approximation guarantees are obtained by comparison with dual feasible solutions found by our algorithm. 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 the maximum weight of a commodity.

Quickest Flows Over Time

A separable assignment problem (SAP) is defined by a set of bins and a set of items to pack in each bin; a value, f ij , for assigning item j to bin i; and a separate packing constraint for each bin - i.e. for bin i, a family L i of subsets of items that fit in bin i. The goal is to pack items into bins to maximize the aggregate value. This class of problems includes the maximum generalized assignment problem (GAP)1) and a distributed caching problem (DCP) described in this paper.Given a β-approximation algorithm for finding the highest value packing of a single bin, we give1. A polynomial-time LP-rounding based ((1 − 1/e)β)-approximation algorithm.2. A simple polynomial-time local search (β/β+1 - e) - approximation algorithm, for any e > 0.Therefore, for all examples of SAP that admit an approximation scheme for the single-bin problem, we obtain an LP-based algorithm with (1 - 1/e - e)-approximation and a local search algorithm with (1/2-e)-approximation guarantee. Furthermore, for cases in which the subproblem admits a fully polynomial approximation scheme (such as for GAP), the LP-based algorithm analysis can be strengthened to give a guarantee of 1 - 1/e. The best previously known approximation algorithm for GAP is a 1/2-approximation by Shmoys and Tardos; and Chekuri and Khanna. Our LP algorithm is based on rounding a new linear programming relaxation, with a provably better integrality gap.To complement these results, we show that SAP and DCP cannot be approximated within a factor better than 1 -1/e unless NP⊆ DTIME(nO(log log n)), even if there exists a polynomial-time exact algorithm for the single-bin problem.We extend the (1 - 1/e)-approximation algorithm to a nonseparable assignment problem with applications in maximizing revenue for budget-constrained combinatorial auctions and the AdWords assignment problem. We generalize the local search algorithm to yield a 1/2-e approximation algorithm for the k-median problem with hard capacities. Finally, we study naturally defined game-theoretic versions of these problems, and show that they have price of anarchy of 2. We also prove the existence of cycles of best response moves, and exponentially long best-response paths to (pure or sink) equilibria.

Iterative rounding 2-approximation algorithms for minimum-cost vertex connectivity problems

We prove the existence of tolls to induce multicommodity, heterogeneous network users that independently choose routes minimizing their own linear function of tolls versus latency to collectively form the traffic pattern of a minimum average latency flow. This generalizes both the previous known results of the existence of tolls for multicommodity, homogeneous users (Beckman et al., 1956) and for single commodity, heterogeneous users (Cole et al., 2003). Unlike previous proofs for single commodity users in general graphs, our proof is constructive - it does not rely on a fixed point theorem - and results in a simple polynomial-sized linear program to compute tolls when the number of different types of users is bounded by a polynomial. We show that our proof gives a complete characterization of flows that are enforceable by tolls. In particular, tolls exist to induce any traffic pattern that is the result of minimizing an arbitrary function from R/sup E(G)/ to the reals that is nondecreasing in each of its arguments. Thus, tolls exist to induce flows with minimum average weighted latency, minimum maximum latency, and other natural objectives. We give an exponential bound on tolls that is independent of the number of network users and the number of commodities. We use this to show that multicommodity tolls also exist when users are not from discrete classes, but instead define a general function that trades off latency versus toll preference. Finally, we show that our result extends to very general frameworks. In particular, we show that tolls exist to induce the Nash equilibrium of general nonatomic congestion games to be system optimal. In particular, tolls exist even when 1) latencies depend on user type; 2) latency functions are nonseparable functions of traffic on edges; 3) the latency of a set S is an arbitrary function of the latencies of the resources contained in S. Our exponential bound on size of tolls also holds in this case; and we give an example of a congestion game that shows this is tight; it requires tolls that are exponential in the size of the game.

On Identifying Strongly Connected Components in Parallel

Flows over time (also called dynamic flows) generalize standard network flows by introducing an element of time. They naturally model problems where travel and transmission are not instantaneous. Traditionally, flows over time are solved in time-expanded networks that contain one copy of the original network for each discrete time step. While this method makes available the whole algorithmic toolbox developed for static flows, its main and often fatal drawback is the enormous size of the time-expanded network. We present several approaches for coping with this difficulty. First, inspired by the work of Ford and Fulkerson on maximal

Scheduling parallelizable tasks to minimize average response time

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A push-relabel framework for submodular function minimization and applications to parametric optimization

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Further Improvements in Competitive Guarantees for QoS Buffering

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