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In this paper, we give the first approximation algorithm for the problem of max-min fair allocation of indivisible goods. An instance of this problem consists of a set of k people and m indivisible goods. Each person has a known linear utility function over the set of goods which might be different from the others’. The goal is to distribute the goods among the people and maximize the minimum utility received by them. 1 The approximation ratio of our algorithm is Ω ( √ k log3). As a crucial part of our k algorithm, we design and analyze an iterative method for rounding a fractional matching on a tree which might be of independent interest. We also provide better bounds when we are allowed to exclude a small fraction of the people from the problem.

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This paper revisits the greedy online load-balancing algorithm for unrelated 1-1 machines from the viewpoint of fairness. We prove that the greedy approach is globally O(log n)-fair where n is the number of jobs. This should be contrasted with polynomial lower bounds presented in [7] for the routing model in which a single job may require multiple resources. We measure fairness via the notion of vector majorization as developed by Hardy, Littlewood, and Polya [13]. Our definition of fairness in terms of approximate majorization is equivalent to the prefix measure proposed by Kleinberg, Rabani, and Tardos [11]. Approximate majorization generalizes the popular notion of max-min fairness to account for the variable allocation of jobs to machines [11, 7]. We also define a machine-centric view of fairness and prove that the greedy online algorithm is globally O(log m)-balanced, where m is the number of machines.

In this work we study a wide range of online and offline routing and packing problems with various objectives. We provide a unified approach, based on a clean primal-dual method, for the design of online algorithms for these problems, as well as improved bounds on the competitive factor. In particular, our analysis uses weak duality rather than a tailor made (i.e., problem specific) potential function. We demonstrate our ideas and results in the context of routing problems. Using our primal-dual approach, we develop a new generic online routing algorithm that outperforms previous algorithms suggested earlier by Azar et al. [5, 4]. We then show the applicability of our generic algorithm to various models and provide improved algorithms for achieving coordinate-wise competitiveness, maximizing throughput, and minimizing maximum load. In particular, we improve the results obtained by Goel et al. [13] by an O(log n) factor for the problem of achieving coordinate-wise competitiveness, and by an O(log log n) factor for the problem of maximizing the throughput. For some of the settings we also prove improved lower bounds. We believe our results further our understanding of the applicability of the primaldual method to online algorithms, and we are confident that the method will prove useful to other online scenarios. Finally, we revisit the notions of coordinate-wise and prefix competitiveness in an offline setting. We design the first polynomial time algorithm that computes an almost optimal coordinate-wise routing for several routing models. We also revisit previously studied routing models [16, 11] and prove tight lower and upper bounds of Θ(log n) on prefix competitiveness for these models. 1

For multi-criteria problems and problems with poorly characterized objective, it is often desirable to simultaneously approximate the optimum solution for a large class of objective functions. We consider two such classes: 1. Maximizing all symmetric concave functions, and 2. Minimizing all symmetric convex functions. The first class corresponds to maximizing profit for a resource allocation problem (such as allocation of bandwidths in a computer network). The concavity requirement corresponds to the law of diminishing returns in economics. The second class corresponds to minimizing cost or congestion in a load balancing problem, where the congestion/cost is some convex function of the loads. Informally, a simultaneous α-approximation for either class is a feasible solution that is within a factor α of the optimum for all functions in that class. Clearly, the structure of the feasible set has a significant impact on the best possible α and the computational complexity of finding a solution that achieves (or nearly achieves) this α. We develop a framework and a set of techniques to perform simultaneous optimization for a wide variety of problems.

A good measure of fairness is an essential prerequisite for a systematic development of fair resource allocation protocols as well as for a systematic evaluation of the fairness of existing protocols. We propose the use of approximate majorization as a framework for quantifying the fairness of a resource allocation scheme. We demonstrate how approximate majorization subsumes and generalizes several natural measures of fairness. We then relate majorization to revenue maximization, and sketch an efficient algorithm to compute the fairness of a given allocation as well to find the fairest allocation. We believe that our framework is quite general and can be applied to several routing, bandwidth allocation, load balancing, and clustering problems. To illustrate the framework in a concrete setting, we perform a preliminary case study of the fairness of TCP as a bandwidth allocation protocol in communication networks, and of multi-path routing vs. single path routing. We discover several interesting trends about the fairness of TCP which merit further study. The fairness of TCP improves as the buffer sizes and/or link capacities are increased. In several realistic cases, the fairness of TCP is comparable to or better than that of max-min fairness. 1

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ABSTRACT. In many optimization problems, a solution can be viewed as ascribing a “cost ” to each client, and the goal is to optimize some aggregation of the per-client costs. We often optimize some Lp-norm (or some other symmetric convex function or norm) of the vector of costs—though different applications may suggest different norms to use. Ideally, we could obtain a solution that optimizes several norms simultaneously. In this paper, we examine approximation algorithms that simultaneously perform well on all norms, or on all Lp norms. A natural problem in this framework is the Lp Set Cover problem, which generalizes SET COVER and MIN-SUM SET COVER. We show that the greedy algorithm simultaneously gives a (p + ln p + O(1))approximation for all p, and show that this approximation ratio is optimal up to constants under reasonable complexity-theoretic assumptions. We additionally show how to use our analysis techniques to give similar results for the more general submodular set cover, and prove some results for the so-called pipelined set cover problem. We then go on to examine approximation algorithms in the “all-norms ” and the “all-Lp-norms ” frameworks more broadly, and present algorithms and structural results for other problems such as k-facility-location, TSP, and average flow-time minimization, extending and unifying previously known results. 1

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