Niv Buchbinder

Online primal-dual algorithms for maximizing ad-auctions revenue

We study the online ad-auctions problem introduced by Mehta et al. [15]. We design a (1 - 1/e)-competitive (optimal) algorithm for the problem, which is based on a clean primal-dual approach, matching the competitive factor obtained in [15]. Our basic algorithm along with its analysis are very simple. Our results are based on a unified approach developed earlier for the design of online algorithms [7,8]. In particular, the analysis uses weak duality rather than a tailor made (i.e., problem specific) potential function. We show that this approach is useful for analyzing other classical online algorithms such as ski rental and the TCP-acknowledgement problem. We are confident that the primal-dual method will prove useful in other online scenarios as well. The primal-dual approach enables us to extend our basic ad-auctions algorithm in a straight forward manner to scenarios in which additional information is available, yielding improved worst case competitive factors. In particular, a scenario in which additional stochastic information is available to the algorithm, a scenario in which the number of interested buyers in each product is bounded by some small number d, and a general risk management framework.

A Tight Linear Time (1/2)-Approximation for Unconstrained Submodular Maximization

We consider the Unconstrained Submodular Maximization problem in which we are given a non-negative submodular function f : 2N → ℝ+, and the objective is to find a subset S ⊆ N maximizing f(S). This is one of the most basic submodular optimization problems, having a wide range of applications. Some well known problems captured by Unconstrained Submodular Maximization include MaxCut, Max-DiCut, and variants of Max-SAT and maximum facility location. We present a simple randomized linear time algorithm achieving a tight approximation guarantee of 1/2, thus matching the known hardness result of Feige et al. [11]. Our algorithm is based on an adaptation of the greedy approach which exploits certain symmetry properties of the problem. Our method might seem counterintuitive, since it is known that the greedy algorithm fails to achieve any bounded approximation factor for the problem.

Online primal-dual algorithms for covering and packing problems

We study a wide range of online covering and packing optimization problems. In an online covering problem a linear cost function is known in advance, but the linear constraints that define the feasible solution space are given one by one in an online fashion. In an online packing problem the profit function as well as the exact packing constraints are not fully known in advance. In each round additional information about the profit function and the constraints is revealed. We provide general deterministic schemes for online fractional covering and packing problems. We also provide deterministic algorithms for a couple of integral covering and packing problems.

A general approach to online network optimization problems

We study a wide range of online graph and network optimization problems, focusing on problems that arise in the study of connectivity and cuts in graphs. In a general online network design problem, we have a communication network known to the algorithm in advance. What is not known in advance are the bandwidth or cut demands between nodes in the network. Our results include an <i>O</i>(log <i>m</i> log <i>n</i>) competitive randomized algorithm for the online non-metric facility location and for a generalization of the problem called themulticast problem. In the non-metric facility location <i>m</i> is the number of facilities and <i>n</i> is the number of clients. The competitive ratio is nearly tight. We also present an<i>O</i>(log² <i>n</i> log <i>k</i>) competitive randomized algorithm for the on-line group Steiner problem in trees and an <i>O</i>(log³ <i>n</i> log <i>k</i>)competitive randomized algorithm for the problem in general graphs, where <i>n</i> is the number of vertices in the graph and <i>k</i> is the number of groups. Finally, we design a deterministic <i>O</i>(log³ <i>n</i> log log <i>n</i>) competitive algorithm for the online multi-cut problem. Our algorithms are based on a unified framework for designing online algorithms for problems involving connectivity and cuts. We first present a general <i>O</i>(log <i>m</i>)-deterministic algorithm for generating fractional solution that satisfies the online connectivity or cut demands, where <i>m</i> is the number of edges in the graph.

Online job-migration for reducing the electricity bill in the cloud

Energy costs are becoming the fastest-growing element in datacenter operation costs. One basic approach to reduce these costs is to exploit the spatiotemporal variation in electricity prices by moving computation to datacenters in which energy is available at a cheaper price. However, injudicious job migration between datacenters might increase the overall operation cost due to the bandwidth costs of transferring application state and data over the wide-area network. To address this challenge, we propose novel online algorithms for migrating batch jobs between datacenters, which handle the fundamental tradeoff between energy and bandwidth costs. A distinctive feature of our algorithms is that they consider not only the current availability and cost of (possibly multiple) energy sources, but also the future variability and uncertainty thereof. Using the framework of competitive-analysis, we establish worst-case performance bounds for our basic online algorithm. We then propose a practical, easy-to-implement version of the basic algorithm, and evaluate it through simulations on real electricity pricing and job workload data. The simulation results indicate that our algorithm outperforms plausible greedy algorithms that ignore future outcomes. Notably, the actual performance of our approach is significantly better than the theoretical guarantees, within 6% of the optimal offline solution.

Online Primal-Dual Algorithms for Covering and Packing

We study a wide range of online covering and packing optimization problems. In an online covering problem, a linear cost function is known in advance, but the linear constraints that define the feasible solution space are given one by one, in rounds. In an online packing problem, the profit function as well as the packing constraints are not known in advance. In each round additional information (i.e., a new variable) about the profit function and the constraints is revealed. An online algorithm needs to maintain a feasible solution in each round; in addition, the solutions generated over the different rounds need to satisfy a monotonicity property. We provide general deterministic primal-dual algorithms for online fractional covering and packing problems. We also provide deterministic algorithms for several integral online covering and packing problems. Our algorithms are designed via a novel online primal-dual technique and are evaluated via competitive analysis.

Improved Bounds for Online Routing and Packing Via a Primal-Dual Approach

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 Y. Azar et al. (1993, 1997). 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 A. Goel et al. (2001) 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 primal-dual 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 by A. Kumar and J.M. Kleinberg (2000) and A. Goel and A. Meyerson (2005) and prove tight lower and upper bounds of Theta(log n) on prefix competitiveness for these models

Randomized competitive algorithms for generalized caching

We consider online algorithms for the generalized caching problem. Here we are given a cache of size k and pages with arbitrary sizes and fetching costs. Given a request sequence of pages, the goal is to minimize the total cost of fetching the pages into the cache. We give an online algorithm with competitive ratio O(log2k), which is the first algorithm for the problem with competitive ratio sublinear in k. We also give improved O(log k)-competitive algorithms for the special cases of the Bit Model and Fault model. In the Bit Model, the fetching cost is proportional to the size of the page and in the Fault model all fetching costs are uniform. Previously, an O(log2 k)-competitive algorithm due to Irani [14] was known for both of these models. Our algorithms are based on an extension of the primal-dual framework for online algorithms which was developed by Buchbinder and Naor [7]. We first generate an O(log k)-competitive fractional algorithm for the problem. This is done by using a strengthened LP formulation with knapsack-cover constraints, where exponentially many constraints are added upon arrival of a new request. Second, we round online the fractional solution and obtain a randomized online algorithm. Our techniques provide a unified framework for caching algorithms and are substantially simpler than those previously used.

A Polylogarithmic-Competitive Algorithm for the k -Server Problem

We give the first polylogarithmic-competitive randomized algorithm for the k-server problem on an arbitrary finite metric space. In particular, our algorithm achieves a competitive ratio of Õ(log3 n log2 k) for any metric space on n points. This improves upon the (2k-1)-competitive algorithm of Koutsoupias and Papadimitriou (J. ACM 1995) whenever n is sub-exponential in k.

Secretary Problems via Linear Programming

In the classical secretary problem an employer would like to choose the best candidate among n competing candidates that arrive in a random order. In each iteration, one candidates rank vis-a-vis previously arrived candidates is revealed and the employer makes an irrevocable decision about her selection. This basic concept of n elements arriving in a random order and irrevocable decisions made by an algorithm have been explored extensively over the years, and used for modeling the behavior of many processes. Our main contribution is a new linear programming technique that we introduce as a tool for obtaining and analyzing algorithms for the secretary problem and its variants. The linear program is formulated using judiciously chosen variables and constraints and we show a one-to-one correspondence between algorithms for the secretary problem and feasible solutions to the linear program. Capturing the set of algorithms as a linear polytope holds the following immediate advantages: Computing the optimal algorithm reduces to solving a linear program. Proving an upper bound on the performance of any algorithm reduces to finding a feasible solution to the dual program. Exploring variants of the problem is as simple as adding new constraints, or manipulating the objective function of the linear program. We demonstrate these ideas by exploring some natural variants of the secretary problem. In particular, using our approach, we design optimal secretary algorithms in which the probability of selecting a candidate at any position is equal. We refer to such algorithms as position independent and these algorithms are motivated by the recent applications of secretary problems to online auctions. We also show a family of linear programs that characterize all algorithms that are allowed to choose J candidates and gain profit from the K best candidates. We believe that a linear programming based approach may be very helpful in the context of other variants of the secretary problem.