arXiv subjects

Yossi Azar

Publications and source records attributed to Yossi Azar.

At least 37 records · Page 2Linked to original sources

arXiv2020

Beyond Tree Embeddings -- a Deterministic Framework for Network Design with Deadlines or Delay

We consider network design problems with deadline or delay. All previous results for these models are based on randomized embedding of the graph into a tree (HST) and then solving the problem on this tree. We show that this is not necessary. In particular, we design a deterministic framework for these problems which is not based on embedding. This enables us to provide deterministic $\text{poly-log}(n)$-competitive algorithms for Steiner tree, generalized Steiner tree, node weighted Steiner tree, (non-uniform) facility location and directed Steiner tree with deadlines or with delay (where $n$ is the number of nodes). Our deterministic algorithms also give improved guarantees over some previous randomized results. In addition, we show a lower bound of $\text{poly-log}(n)$ for some of these problems, which implies that our framework is optimal up to the power of the poly-log. Our algorithms and techniques differ significantly from those in all previous considerations of these problems.

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arXiv2019

General Framework for Metric Optimization Problems with Delay or with Deadlines

In this paper, we present a framework used to construct and analyze algorithms for online optimization problems with deadlines or with delay over a metric space. Using this framework, we present algorithms for several different problems. We present an $O(D^{2})$-competitive deterministic algorithm for online multilevel aggregation with delay on a tree of depth $D$, an exponential improvement over the $O(D^{4}2^{D})$-competitive algorithm of Bienkowski et al. (ESA '16), where the only previously-known improvement was for the special case of deadlines by Buchbinder et al. (SODA '17). We also present an $O(\log^{2}n)$-competitive randomized algorithm for online service with delay over any general metric space of $n$ points, improving upon the $O(\log^{4}n)$-competitive algorithm by Azar et al. (STOC '17). In addition, we present the problem of online facility location with deadlines. In this problem, requests arrive over time in a metric space, and need to be served until their deadlines by facilities that are opened momentarily for some cost. We also consider the problem of facility location with delay, in which the deadlines are replaced with arbitrary delay functions. For those problems, we present $O(\log^{2}n)$-competitive algorithms, with $n$ the number of points in the metric space. The algorithmic framework we present includes techniques for the design of algorithms as well as techniques for their analysis.

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arXiv2018

Improved Online Algorithm for Weighted Flow Time

We discuss one of the most fundamental scheduling problem of processing jobs on a single machine to minimize the weighted flow time (weighted response time). Our main result is a $O(\log P)$-competitive algorithm, where $P$ is the maximum-to-minimum processing time ratio, improving upon the $O(\log^{2}P)$-competitive algorithm of Chekuri, Khanna and Zhu (STOC 2001). We also design a $O(\log D)$-competitive algorithm, where $D$ is the maximum-to-minimum density ratio of jobs. Finally, we show how to combine these results with the result of Bansal and Dhamdhere (SODA 2003) to achieve a $O(\log(\min(P,D,W)))$-competitive algorithm (where $W$ is the maximum-to-minimum weight ratio), without knowing $P,D,W$ in advance. As shown by Bansal and Chan (SODA 2009), no constant-competitive algorithm is achievable for this problem.

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arXiv2018

Deterministic Min-Cost Matching with Delays

We consider the online Minimum-Cost Perfect Matching with Delays (MPMD) problem introduced by Emek et al. (STOC 2016), in which a general metric space is given, and requests are submitted in different times in this space by an adversary. The goal is to match requests, while minimizing the sum of distances between matched pairs in addition to the time intervals passed from the moment each request appeared until it is matched. In the online Minimum-Cost Bipartite Perfect Matching with Delays (MBPMD) problem introduced by Ashlagi et al. (APPROX/RANDOM 2017), each request is also associated with one of two classes, and requests can only be matched with requests of the other class. Previous algorithms for the problems mentioned above, include randomized $O\left(\log n\right)$-competitive algorithms for known and finite metric spaces, $n$ being the size of the metric space, and a deterministic $O\left(m\right)$-competitive algorithm, $m$ being the number of requests. We introduce $O\left(m^{\log\left(\frac{3}{2}+ε\right)}\right)$-competitive deterministic algorithms for both problems and for any fixed $ε> 0$. In particular, for a small enough $ε$ the competitive ratio becomes $O\left(m^{0.59}\right)$. These are the first deterministic algorithms for the mentioned online matching problems, achieving a sub-linear competitive ratio. Our algorithms do not need to know the metric space in advance.

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arXiv2017

Prophet Secretary: Surpassing the $1-1/e$ Barrier

In the Prophet Secretary problem, samples from a known set of probability distributions arrive one by one in a uniformly random order, and an algorithm must irrevocably pick one of the samples as soon as it arrives. The goal is to maximize the expected value of the sample picked relative to the expected maximum of the distributions. This is one of the most simple and fundamental problems in online decision making that models the process selling one item to a sequence of costumers. For a closely related problem called the Prophet Inequality where the order of the random variables is adversarial, it is known that one can achieve in expectation $1/2$ of the expected maximum, and no better ratio is possible. For the Prophet Secretary problem, that is, when the variables arrive in a random order, Esfandiari et al.\ (ESA 2015) showed that one can actually get $1-1/e$ of the maximum. The $1-1/e$ bound was recently extended to more general settings (Ehsani et al., 2017). Given these results, one might be tempted to believe that $1-1/e$ is the correct bound. We show that this is not the case by providing an algorithm for the Prophet Secretary problem that beats the $1-1/e$ bound and achieves $1-1/e+1/400$ of the optimum value. We also prove a hardness result on the performance of algorithms under a natural restriction which we call deterministic distribution-insensitivity.

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arXiv2017

The Strategy of Experts for Repeated Predictions

We investigate the behavior of experts who seek to make predictions with maximum impact on an audience. At a known future time, a certain continuous random variable will be realized. A public prediction gradually converges to the outcome, and an expert has access to a more accurate prediction. We study when the expert should reveal his information, when his reward is based on a proper scoring rule (e.g., is proportional to the change in log-likelihood of the outcome). In Azar et. al. (2016), we analyzed the case where the expert may make a single prediction. In this paper, we analyze the case where the expert is allowed to revise previous predictions. This leads to a rather different set of dilemmas for the strategic expert. We find that it is optimal for the expert to always tell the truth, and to make a new prediction whenever he has a new signal. We characterize the expert's expectation for his total reward, and show asymptotic limits

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arXiv2017

Online Service with Delay

In this paper, we introduce the online service with delay problem. In this problem, there are $n$ points in a metric space that issue service requests over time, and a server that serves these requests. The goal is to minimize the sum of distance traveled by the server and the total delay in serving the requests. This problem models the fundamental tradeoff between batching requests to improve locality and reducing delay to improve response time, that has many applications in operations management, operating systems, logistics, supply chain management, and scheduling. Our main result is to show a poly-logarithmic competitive ratio for the online service with delay problem. This result is obtained by an algorithm that we call the preemptive service algorithm. The salient feature of this algorithm is a process called preemptive service, which uses a novel combination of (recursive) time forwarding and spatial exploration on a metric space. We hope this technique will be useful for related problems such as reordering buffer management, online TSP, vehicle routing, etc. We also generalize our results to $k > 1$ servers.

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arXiv2016

Polylogarithmic Bounds on the Competitiveness of Min-cost (Bipartite) Perfect Matching with Delays

We consider the problem of online Min-cost Perfect Matching with Delays (MPMD) recently introduced by Emek et al, (STOC 2016). This problem is defined on an underlying $n$-point metric space. An adversary presents real-time requests online at points of the metric space, and the algorithm is required to match them, possibly after keeping them waiting for some time. The cost incurred is the sum of the distances between matched pairs of points (the connection cost), and the sum of the waiting times of the requests (the delay cost). We present an algorithm with a competitive ratio of $O(\log n)$, which improves the upper bound of $O(\log^2n+\logΔ)$ of Emek et al, by removing the dependence on $Δ$, the aspect ratio of the metric space (which can be unbounded as a function of $n$). The core of our algorithm is a deterministic algorithm for MPMD on metrics induced by edge-weighted trees of height $h$, whose cost is guaranteed to be at most $O(1)$ times the connection cost plus $O(h)$ times the delay cost of every feasible solution. The reduction from MPMD on arbitrary metrics to MPMD on trees is achieved using the result on embedding $n$-point metric spaces into distributions over weighted hierarchically separated trees of height $O(\log n)$, with distortion $O(\log n)$. We also prove a lower bound of $Ω(\sqrt{\log n})$ on the competitive ratio of any randomized algorithm. This is the first lower bound which increases with $n$, and is attained on the metric of $n$ equally spaced points on a line. The problem of Min-cost Bipartite Perfect Matching with Delays (MBPMD) is the same as MPMD except that every request is either positive or negative, and requests can be matched only if they have opposite polarity. We prove an upper bound of $O(\log n)$ and a lower bound of $Ω(\log^{1/3}n)$ on the competitive ratio of MBPMD with a more involved analysis.

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arXiv2016

When should an expert make a prediction?

We consider a setting where in a known future time, a certain continuous random variable will be realized. There is a public prediction that gradually converges to its realized value, and an expert that has access to a more accurate prediction. Our goal is to study {\em when} should the expert reveal his information, assuming that his reward is based on a logarithmic market scoring rule (i.e., his reward is proportional to the gain in log-likelihood of the realized value). Our contributions are: (1) we characterize the expert's optimal policy and show that it is threshold based. (2) we analyze the expert's asymptotic expected optimal reward and show a tight connection to the Law of the Iterated Logarithm, and (3) we give an efficient dynamic programming algorithm to compute the optimal policy.

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arXiv2016

Online Lower Bounds via Duality

In this paper, we exploit linear programming duality in the online setting (i.e., where input arrives on the fly) from the unique perspective of designing lower bounds on the competitive ratio. In particular, we provide a general technique for obtaining online deterministic and randomized lower bounds (i.e., hardness results) on the competitive ratio for a wide variety of problems. We show the usefulness of our approach by providing new, tight lower bounds for three diverse online problems. The three problems we show tight lower bounds for are the Vector Bin Packing problem, Ad-auctions (and various online matching problems), and the Capital Investment problem. Our methods are sufficiently general that they can also be used to reconstruct existing lower bounds. Our techniques are in stark contrast to previous works, which exploit linear programming duality to obtain positive results, often via the useful primal-dual scheme. We design a general recipe with the opposite aim of obtaining negative results via duality. The general idea behind our approach is to construct a primal linear program based on a collection of input sequences, where the objective function corresponds to optimizing the competitive ratio. We then obtain the corresponding dual linear program and provide a feasible solution, where the objective function yields a lower bound on the competitive ratio. Online lower bounds are often achieved by adapting the input sequence according to an online algorithm's behavior and doing an appropriate ad hoc case analysis. Using our unifying techniques, we simultaneously combine these cases into one linear program and achieve online lower bounds via a more robust analysis. We are confident that our framework can be successfully applied to produce many more lower bounds for a wide array of online problems.

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arXiv2015

Liquid Price of Anarchy

Incorporating budget constraints into the analysis of auctions has become increasingly important, as they model practical settings more accurately. The social welfare function, which is the standard measure of efficiency in auctions, is inadequate for settings with budgets, since there may be a large disconnect between the value a bidder derives from obtaining an item and what can be liquidated from her. The Liquid Welfare objective function has been suggested as a natural alternative for settings with budgets. Simple auctions, like simultaneous item auctions, are evaluated by their performance at equilibrium using the Price of Anarchy (PoA) measure -- the ratio of the objective function value of the optimal outcome to the worst equilibrium. Accordingly, we evaluate the performance of simultaneous item auctions in budgeted settings by the Liquid Price of Anarchy (LPoA) measure -- the ratio of the optimal Liquid Welfare to the Liquid Welfare obtained in the worst equilibrium. Our main result is that the LPoA for mixed Nash equilibria is bounded by a constant when bidders are additive and items can be divided into sufficiently many discrete parts. Our proofs are robust, and can be extended to achieve similar bounds for simultaneous second price auctions as well as Bayesian Nash equilibria. For pure Nash equilibria, we establish tight bounds on the LPoA for the larger class of fractionally-subadditive valuations. To derive our results, we develop a new technique in which some bidders deviate (surprisingly) toward a non-optimal solution. In particular, this technique does not fit into the smoothness framework.

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arXiv2015

Truthful Online Scheduling with Commitments

We study online mechanisms for preemptive scheduling with deadlines, with the goal of maximizing the total value of completed jobs. This problem is fundamental to deadline-aware cloud scheduling, but there are strong lower bounds even for the algorithmic problem without incentive constraints. However, these lower bounds can be circumvented under the natural assumption of deadline slackness, i.e., that there is a guaranteed lower bound $s > 1$ on the ratio between a job's size and the time window in which it can be executed. In this paper, we construct a truthful scheduling mechanism with a constant competitive ratio, given slackness $s > 1$. Furthermore, we show that if $s$ is large enough then we can construct a mechanism that also satisfies a commitment property: it can be determined whether or not a job will finish, and the requisite payment if so, well in advance of each job's deadline. This is notable because, in practice, users with strict deadlines may find it unacceptable to discover only very close to their deadline that their job has been rejected.

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arXiv2015

TSP with Time Windows and Service Time

We consider TSP with time windows and service time. In this problem we receive a sequence of requests for a service at nodes in a metric space and a time window for each request. The goal of the online algorithm is to maximize the number of requests served during their time window. The time to traverse an edge is the distance between the incident nodes of that edge. Serving a request requires unit time. We characterize the competitive ratio for each metric space separately. The competitive ratio depends on the relation between the minimum laxity (the minimum length of a time window) and the diameter of the metric space. Specifically, there is a constant competitive algorithm depending whether the laxity is larger or smaller than the diameter. In addition, we characterize the rate of convergence of the competitive ratio to $1$ as the laxity increases. Specifically, we provide a matching lower and upper bounds depending on the ratio between the laxity and the TSP of the metric space (the minimum distance to traverse all nodes). An application of our result improves the lower bound for colored packets with transition cost and matches the upper bound. In proving our lower bounds we use an interesting non-standard embedding with some special properties. This embedding may be interesting by its own.

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arXiv2014

Online Covering with Convex Objectives and Applications

We give an algorithmic framework for minimizing general convex objectives (that are differentiable and monotone non-decreasing) over a set of covering constraints that arrive online. This substantially extends previous work on online covering for linear objectives (Alon {\em et al.}, STOC 2003) and online covering with offline packing constraints (Azar {\em et al.}, SODA 2013). To the best of our knowledge, this is the first result in online optimization for generic non-linear objectives; special cases of such objectives have previously been considered, particularly for energy minimization. As a specific problem in this genre, we consider the unrelated machine scheduling problem with startup costs and arbitrary $\ell_p$ norms on machine loads (including the surprisingly non-trivial $\ell_1$ norm representing total machine load). This problem was studied earlier for the makespan norm in both the offline (Khuller~{\em et al.}, SODA 2010; Li and Khuller, SODA 2011) and online settings (Azar {\em et al.}, SODA 2013). We adapt the two-phase approach of obtaining a fractional solution and then rounding it online (used successfully to many linear objectives) to the non-linear objective. The fractional algorithm uses ideas from our general framework that we described above (but does not fit the framework exactly because of non-positive entries in the constraint matrix). The rounding algorithm uses ideas from offline rounding of LPs with non-linear objectives (Azar and Epstein, STOC 2005; Kumar {\em et al.}, FOCS 2005). Our competitive ratio is tight up to a logarithmic factor. Finally, for the important special case of total load ($\ell_1$ norm), we give a different rounding algorithm that obtains a better competitive ratio than the generic rounding algorithm for $\ell_p$ norms. We show that this competitive ratio is asymptotically tight.

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arXiv2013

Colored Packets with Deadlines and Metric Space Transition Cost

We consider scheduling of colored packets with transition costs which form a general metric space. We design a $1 - O(\sqrt{MST(G) / L})$ competitive algorithm. Our main result is a hardness result of $1 - Ω(\sqrt{MST(G) / L})$ which matches the competitive ratio of the algorithm for each metric space separately. In particular, we improve the hardness result of Azar at el. 2009 for uniform metric spaces. We also extend our result to weighted directed graphs which obey the triangular inequality and show a $1 - O(\sqrt{TSP(G) / L})$ competitive algorithm and a nearly-matching hardness result. In proving our hardness results we use some interesting non-standard embedding.

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arXiv2012

Efficient Submodular Function Maximization under Linear Packing Constraints

We study the problem of maximizing a monotone submodular set function subject to linear packing constraints. An instance of this problem consists of a matrix $A \in [0,1]^{m \times n}$, a vector $b \in [1,\infty)^m$, and a monotone submodular set function $f: 2^{[n]} \rightarrow \bbR_+$. The objective is to find a set $S$ that maximizes $f(S)$ subject to $A x_{S} \leq b$, where $x_S$ stands for the characteristic vector of the set $S$. A well-studied special case of this problem is when $f$ is linear. This special case captures the class of packing integer programs. Our main contribution is an efficient combinatorial algorithm that achieves an approximation ratio of $Ω(1 / m^{1/W})$, where $W = \min\{b_i / A_{ij} : A_{ij} > 0\}$ is the width of the packing constraints. This result matches the best known performance guarantee for the linear case. One immediate corollary of this result is that the algorithm under consideration achieves constant factor approximation when the number of constraints is constant or when the width of the constraints is sufficiently large. This motivates us to study the large width setting, trying to determine its exact approximability. We develop an algorithm that has an approximation ratio of $(1 - ε)(1 - 1/e)$ when $W = Ω(\ln m / ε^2)$. This result essentially matches the theoretical lower bound of $1 - 1/e$. We also study the special setting in which the matrix $A$ is binary and $k$-column sparse. A $k$-column sparse matrix has at most $k$ non-zero entries in each of its column. We design a fast combinatorial algorithm that achieves an approximation ratio of $Ω(1 / (Wk^{1/W}))$, that is, its performance guarantee only depends on the sparsity and width parameters.

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arXiv2012

Online Load Balancing on Unrelated Machines with Startup Costs

Motivated by applications in energy-efficient scheduling in data centers, Khuller, Li, and Saha introduced the {\em machine activation} problem as a generalization of the classical optimization problems of set cover and load balancing on unrelated machines. In this problem, a set of $n$ jobs have to be distributed among a set of $m$ (unrelated) machines, given the processing time of each job on each machine, where each machine has a startup cost. The goal is to produce a schedule of minimum total startup cost subject to a constraint $\bf L$ on its makespan. While Khuller {\em et al} considered the offline version of this problem, a typical scenario in scheduling is one where jobs arrive online and have to be assigned to a machine immediately on arrival. We give an $(O(\log (mn)\log m), O(\log m))$-competitive randomized online algorithm for this problem, i.e. the schedule produced by our algorithm has a makespan of $O({\bf L} \log m)$ with high probability, and a total expected startup cost of $O(\log (mn)\log m)$ times that of an optimal offline schedule with makespan $\bf L$. The competitive ratios of our algorithm are (almost) optimal. Our algorithms use the online primal dual framework introduced by Alon {\em et al} for the online set cover problem, and subsequently developed further by Buchbinder, Naor, and co-authors. To the best of our knowledge, all previous applications of this framework have been to linear programs (LPs) with either packing or covering constraints. One novelty of our application is that we use this framework for a mixed LP that has both covering and packing constraints. We hope that the algorithmic techniques developed in this paper to simultaneously handle packing and covering constraints will be useful for solving other online optimization problems as well.

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arXiv2010

Ranking with Submodular Valuations

We study the problem of ranking with submodular valuations. An instance of this problem consists of a ground set $[m]$, and a collection of $n$ monotone submodular set functions $f^1, \ldots, f^n$, where each $f^i: 2^{[m]} \to R_+$. An additional ingredient of the input is a weight vector $w \in R_+^n$. The objective is to find a linear ordering of the ground set elements that minimizes the weighted cover time of the functions. The cover time of a function is the minimal number of elements in the prefix of the linear ordering that form a set whose corresponding function value is greater than a unit threshold value. Our main contribution is an $O(\ln(1 / ε))$-approximation algorithm for the problem, where $ε$ is the smallest non-zero marginal value that any function may gain from some element. Our algorithm orders the elements using an adaptive residual updates scheme, which may be of independent interest. We also prove that the problem is $Ω(\ln(1 / ε))$-hard to approximate, unless P = NP. This implies that the outcome of our algorithm is optimal up to constant factors.

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