$k$-means++ and the related greedy $k$-means++ algorithm are celebrated algorithms that efficiently compute seeds for Lloyd's algorithm. Greedy $k$-means++ is a generalization of $k$-means++ where, in each iteration, a new seed is greedily chosen among multiple $\ell \geq 2$ points sampled, as opposed to a single seed being sampled in $k$-means++. While empirical studies consistently show the superior performance of greedy $k$-means++, making it a preferred method in practice, a discrepancy exists between theory and practice. No theoretical justification currently explains this improved performance. Indeed, the prevailing theory suggests that greedy $k$-means++ exhibits worse performance than $k$-means++ in worst-case scenarios. This paper presents an analysis demonstrating the outperformance of the greedy algorithm compared to $k$-means++ for a natural class of well-separated instances with exponentially decaying distributions, such as Gaussian, specifically when $\ell = \Theta(\log k)$, a common parameter setting in practical applications.

In this paper, we explore how a natural generalization of Shortest Remaining Processing Time (SRPT) can be a powerful meta-algorithm for online scheduling. The meta-algorithm processes jobs to maximally reduce the objective of the corresponding offline scheduling problem of the remaining jobs: minimizing the total weighted completion time of them (the residual optimum). We show that it achieves scalability for minimizing total weighted flow time when the residual optimum exhibits supermodularity. Scalability here means it is O (1)-competitive with an arbitrarily small speed augmentation advantage over the adversary, representing the best possible outcome achievable for various scheduling problems.

Cardinality estimation and conjunctive query evaluation are two of the most fundamental problems in database query processing. Recent work proposed, studied, and implemented a robust and practical information-theoretic cardinality estimation framework. In this framework, the estimator is the cardinality upper bound of a conjunctive query subject to ''degree-constraints'', which model a rich set of input data statistics. For general degree constraints, computing this bound is computationally hard. Researchers have naturally sought efficiently computable relaxed upper bounds that are as tight as possible. The polymatroid bound is the tightest among those relaxed upper bounds. While it is an open question whether the polymatroid bound can be computed in polynomial-time in general, it is known to be computable in polynomial-time for some classes of degree constraints. Our focus is on a common class of degree constraints called simple degree constraints. Researchers had not previously determined how to compute the polymatroid bound in polynomial time for this class of constraints. Our first main result is a polynomial time algorithm to compute the polymatroid bound given simple degree constraints. Our second main result is a polynomial-time algorithm to compute a ''proof sequence'' establishing this bound. This proof sequence can then be incorporated in the PANDA-framework to give a faster algorithm to evaluate a conjunctive query. In addition, we show computational limitations to extending our results to broader classes of degree constraints. Finally, our technique leads naturally to a new relaxed upper bound called the flow bound, which is computationally tractable.

The feedback arc set problem is one of the most fundamental and well-studied ranking problems where n objects are to be ordered based on their pairwise comparison. The problem enjoys several efficient approximation algorithms in the offline setting. Unfortunately, online there are strong lower bounds on the competitive ratio establishing that no algorithm can perform well in the worst case. This paper introduces a new beyond-worst-case model for online feedback arc set. In the model, a sample of the input is given to the algorithm offline before the remaining instance is revealed online. This models the case in practice where yesterday's data is available and is similar to today's online instance. This sample is drawn from a known distribution which may not be uniform. We design an online algorithm with strong theoretical guarantees. The algorithm has a small constant competitive ratio when the sample is uniform---if not, we show we can recover the same result by adding a provably minimal sample. Empirical results validate the theory and show that such algorithms can be used on temporal data to obtain strong results.

Algorithms with (machine-learned) predictions is a powerful framework for combining traditional worst-case algorithms with modern machine learning. However, the vast majority of work in this space assumes that the prediction itself is non-probabilistic, even if it is generated by some stochastic process (such as a machine learning system). This is a poor fit for modern ML, particularly modern neural networks, which naturally generate a *distribution*. We initiate the study of algorithms with *distributional* predictions, where the prediction itself is a distribution. We focus on one of the simplest yet fundamental settings: binary search (or searching a sorted array). This setting has one of the simplest algorithms with a point prediction, but what happens if the prediction is a distribution? We show that this is a richer setting: there are simple distributions where using the classical prediction-based algorithm with any single prediction does poorly. Motivated by this, as our main result, we give an algorithm with query complexity $O(H(p) + \log \eta)$, where $H(p)$ is the entropy of the true distribution $p$ and $\eta$ is the earth mover's distance between $p$ and the predicted distribution $\hat p$. This also yields the first *distributionally-robust* algorithm for the classical problem of computing an optimal binary search tree given a distribution over target keys. We complement this with a lower bound showing that this query complexity is essentially optimal (up to constants), and experiments validating the practical usefulness of our algorithm.

The classical ski-rental problem admits a textbook 2-competitive deterministic algorithm, and a simple randomized algorithm that is e/e-1-competitive in expectation. The randomized algorithm, while optimal in expectation, has a large variance in its performance: it has more than a 37% chance of competitive ratio exceeding 2, and the change of the competitive ratio exceeding n is Θ(1/n)!

Online load balancing for heterogeneous machines aims to minimize the makespan (maximum machine workload) by scheduling arriving jobs with varying sizes on different machines. In the adversarial setting, where an adversary chooses not only the collection of job sizes but also their arrival order, the problem is well-understood and the optimal competitive ratio is known to be Θ(log m) where m is the number of machines. In the more realistic random arrival order model, the understanding is limited. Previously, the best lower bound on the competitive ratio was only Ω(log log m).

This paper explores the design of a balanced data-sharing marketplace for entities with heterogeneous datasets and machine learning models that they seek to refine using data from other agents. The goal of the marketplace is to encourage participation for data sharing in the presence of such heterogeneity. Our market design approach for data sharing focuses on interim utility balance, where participants contribute and receive equitable utility from refinement of their models. We present such a market model for which we study computational complexity, solution existence, and approximation algorithms for welfare maximization and core stability. We finally support our theoretical insights with simulations on a mean estimation task inspired by road traffic delay estimation.

Datalog o is an extension of Datalog that allows for aggregation and recursion over an arbitrary commutative semiring. Like Datalog, Datalogo programs can be evaluated via the natural iterative algorithm until a fixed point is reached. However unlike Datalog, the natural iterative evaluation of some Datalogo programs over some semirings may not converge. It is known that the commutative semirings for which the iterative evaluation of Datalogo programs is guaranteed to converge are exactly those semirings that are stable. Previously, the best known upper bound on the number of iterations until convergence over p-stable semirings is ∑i=1 ^n (p+2) i = Θ(p n ) steps, where n is (essentially) the output size. We establish that, in fact, the natural iterative evaluation of a Datalogo program over a p-stable semiring converges within a polynomial number of iterations. In particular our upper bound is O(σ p n 2 ( n 2 lg Λ + lg σ)) where σ is the number of elements in the semiring present in either the input databases or the Datalogo program, and λ is the maximum number of terms in any product in the Datalogo program.

In the submodular ranking (SR) problem, the input consists of a set of submodular functions defined on a ground set of elements. The goal is to order elements for all the functions to have value above a certain threshold as soon on average as possible, assuming we choose one element per time. The problem is flexible enough to capture various applications in machine learning, including decision trees. This paper considers the min-max version of SR where multiple instances share the ground set. With the view of each instance being associated with an agent, the min-max problem is to order the common elements to minimize the maximum objective of all agents---thus, finding a fair solution for all agents. We give approximation algorithms for this problem and demonstrate their effectiveness in the application of finding a decision tree for multiple agents.

We consider the classic online learning and stochastic multi-armed bandit (MAB) problems, when at each step, the online policy can probe and find out which of a small number ($k$) of choices has better reward (or loss) before making its choice. In this model, we derive algorithms whose regret bounds have exponentially better dependence on the time horizon compared to the classic regret bounds. In particular, we show that probing with $k=2$ suffices to achieve time-independent regret bounds for online linear and convex optimization. The same number of probes improve the regret bound of stochastic MAB with independent arms from $O(\sqrt{nT})$ to $O(n^2 \log T)$, where $n$ is the number of arms and $T$ is the horizon length. For stochastic MAB, we also consider a stronger model where a probe reveals the reward values of the probed arms, and show that in this case, $k=3$ probes suffice to achieve parameter-independent constant regret, $O(n^2)$. Such regret bounds cannot be achieved even with full feedback after the play, showcasing the power of limited ``advice'' via probing before making the play. We also present extensions to the setting where the hints can be imperfect, and to the case of stochastic MAB where the rewards of the arms can be correlated.

We revisit two well-studied scheduling problems in the unrelated machines setting where each job can have a different processing time on each machine. For minimizing total weighted completion time we give a 1.45-approximation, which improves upon the previous 1.488-approximation [Im and Shadloo SODA 2020]. The key technical ingredient in this improvement lies in a new rounding scheme that gives strong negative correlation with less restrictions. For minimizing Lk-norms of machine loads, inspired by [Kalaitzis et al. SODA 2017], we give better approximation algorithms. In particular we give a -approximation for the L2-norm which improves upon the former -approximations due to [Azar-Epstein STOC 2005] and [Kumar et al. JACM 2009].

Learning-augmented algorithms—in which, traditional algorithms are augmented with machine-learned predictions—have emerged as a framework to go beyond worst-case analysis. The overarching goal is to design algorithms that perform near-optimally when the predictions are accurate yet retain certain worst-case guarantees irrespective of the accuracy of the predictions. This framework has been successfully applied to online problems such as caching where the predictions can be used to alleviate uncertainties. In this paper we introduce and study the setting in which the learning-augmented algorithm can utilize the predictions parsimoniously. We consider the caching problem—which has been extensively studied in the learning-augmented setting—and show that one can achieve quantitatively similar results but only using a sublinear number of predictions.

The research area of algorithms with predictions has seen recent success showing how to incorporate machine learning into algorithm design to improve performance when the predictions are correct, while retaining worst-case guarantees when they are not. Most previous work has assumed that the algorithm has access to a single predictor. However, in practice, there are many machine learning methods available, often with incomparable generalization guarantees, making it hard to pick a best method a priori. In this work we consider scenarios where multiple predictors are available to the algorithm and the question is how to best utilize them. Ideally, we would like the algorithm's performance to depend on the quality of the {\em best} predictor. However, utilizing more predictions comes with a cost, since we now have to identify which prediction is best. We study the use of multiple predictors for a number of fundamental problems, including matching, load balancing, and non-clairvoyant scheduling, which have been well-studied in the single predictor setting. For each of these problems we introduce new algorithms that take advantage of multiple predictors, and prove bounds on the resulting performance.

We consider the problem of efficiently estimating the size of the join of a collection of preprocessed relational tables from the perspective of instance optimality analysis. The running time of instance optimal algorithms is comparable to the minimum time needed to verify the correctness of a solution. Previously, instance optimal algorithms were only known when the size of the join was small (as one component of their running time was linear in the join size). We give an instance optimal algorithm for estimating the join size for all instances, including when the join size is large, by removing the dependency on the join size. As a byproduct, we show how to sample rows from the join uniformly at random in a comparable amount of time.

We consider the Matrix Tree Multiplication problem. This problem is a generalization of the classic Matrix Chain Multiplication problem covered in the dynamic programming chapter of many introductory algorithms textbooks. An instance of the Matrix Tree Multiplication problem consists of a rooted tree with a matrix associated with each edge. The output is, for each leaf in the tree, the product of the matrices on the chain/path from the root to that leaf. Matrix multiplications that are shared between various chains need only be computed once, potentially being shared between different root to leaf chains. Algorithms are evaluated by the number of scalar multiplications performed. Our main result is a linear time algorithm for which the number of scalar multiplications performed is at most 15 times the optimal number of scalar multiplications.

A recent line of research investigates how algorithms can be augmented with machine-learned predictions to overcome worst case lower bounds. This area has revealed interesting algorithmic insights into problems, with particular success in the design of competitive online algorithms. However, the question of improving algorithm running times with predictions has largely been unexplored. We take a first step in this direction by combining the idea of machine-learned predictions with the idea of ``warm-starting" primal-dual algorithms. We consider one of the most important primitives in combinatorial optimization: weighted bipartite matching and its generalization to $b$-matching. We identify three key challenges when using learned dual variables in a primal-dual algorithm. First, predicted duals may be infeasible, so we give an algorithm that efficiently maps predicted infeasible duals to nearby feasible solutions. Second, once the duals are feasible, they may not be optimal, so we show that they can be used to quickly find an optimal solution. Finally, such predictions are useful only if they can be learned, so we show that the problem of learning duals for matching has low sample complexity. We validate our theoretical findings through experiments on both real and synthetic data. As a result we give a rigorous, practical, and empirically effective method to compute bipartite matchings.

There has been recent interest in using machine-learned predictions to improve the worst-case guarantees of online algorithms. In this paper we continue this line of work by studying the online knapsack problem, but with very weak predictions: in the form of knowing an upper and lower bound for the number of items of each value. We systematically derive online algorithms that attain the best possible competitive ratio for any fixed prediction; we also extend the results to more general settings such as generalized one-way trading and two-stage online knapsack. Our work shows that even seemingly weak predictions can be utilized effectively to provably improve the performance of online algorithms.

In the single-machine non-clairvoyant scheduling problem, the goal is to minimize the total completion time of jobs whose processing times are unknown a priori. We revisit this well-studied problem and consider the question of how to effectively use (possibly erroneous) predictions of the processing times. We study this question from ground zero by first asking what constitutes a good prediction; we then propose a new measure to gauge prediction quality and design scheduling algorithms with strong guarantees under this measure. Our approach to derive a prediction error measure based on natural desiderata could find applications for other online problems.

This paper considers k-means clustering in the presence of noise. It is known that k-means clustering is highly sensitive to noise, and thus noise should be removed to obtain a quality solution. A popular formulation of this problem is called k-means clustering with outliers. The goal of k-means clustering with outliers is to discard up to a specified number z of points as noise/outliers and then find a k-means solution on the remaining data. The problem has received significant attention, yet current algorithms with theoretical guarantees suffer from either high running time or inherent loss in the solution quality. The main contribution of this paper is two-fold. Firstly, we develop a simple greedy algorithm that has provably strong worst case guarantees. The greedy algorithm adds a simple preprocessing step to remove noise, which can be combined with any k-means clustering algorithm. This algorithm gives the first pseudo-approximation-preserving reduction from k-means with outliers to k-means without outliers. Secondly, we show how to construct a coreset of size O(k log n). When combined with our greedy algorithm, we obtain a scalable, near linear time algorithm. The theoretical contributions are verified experimentally by demonstrating that the algorithm quickly removes noise and obtains a high-quality clustering.

We design and mathematically analyze sampling-based algorithms for regularized loss minimization problems that are implementable in popular computational models for large data, in which the access to the data is restricted in some way. Our main result is that if the regularizer’s effect does not become negligible as the norm of the hypothesis scales, and as the data scales, then a uniform sample of modest size is with high probability a coreset. In the case that the loss function is either logistic regression or soft-margin support vector machines, and the regularizer is one of the common recommended choices, this result implies that a uniform sample of size $O(d \sqrt{n})$ is with high probability a coreset of $n$ points in $\Re^d$. We contrast this upper bound with two lower bounds. The first lower bound shows that our analysis of uniform sampling is tight; that is, a smaller uniform sample will likely not be a core set. The second lower bound shows that in some sense uniform sampling is close to optimal, as significantly smaller core sets do not generally exist.

In this paper, we consider the problem of assigning 2-dimensional vector jobs to identical machines online so to minimize the maximum load on any dimension of any machine. For arbitrary number of dimensions d, this problem is known as vector scheduling, and recent research has established the optimal competitive ratio as O((log d)/(log log d)) (Im et al. FOCS 2015, Azar et al. SODA 2018). But, these results do not shed light on the situation for small number of dimensions, particularly for d = 2 which is of practical interest. In this case, a trivial analysis shows that the classic list scheduling greedy algorithm has a competitive ratio of 3. We show the following improvements over this baseline in this paper: - We give an improved, and tight, analysis of the list scheduling algorithm establishing a competitive ratio of 8/3 for two dimensions. - If the value of opt is known, we improve the competitive ratio to 9/4 using a variant of the classic best fit algorithm for two dimensions. - For any fixed number of dimensions, we design an algorithm that is provably the best possible against a fractional optimum solution. This algorithm provides a proof of concept that we can simulate the optimal algorithm online up to the integrality gap of the natural LP relaxation of the problem.

We give a 1.488-approximation for the classic scheduling problem of minimizing total weighted completion time on unrelated machines. This is a considerable improvement on the recent breakthrough of (1.5 – 10−7)-approximation (STOC 2016, Bansal-Srinivasan-Svensson) and the follow-up result of (1.5 – 1/6000)-approximation (FOCS 2017, Li). Bansal et al. introduced a novel rounding scheme yielding strong negative correlations for the first time and applied it to the scheduling problem to obtain their breakthrough, which resolved the open problem if one can beat out the long-standing 1.5-approximation barrier based on independent rounding. Our key technical contribution is in achieving significantly stronger negative correlations via iterative fair contention resolution, which is of independent interest. Previously, Bansal et al. obtained strong negative correlations via a variant of pipage type rounding and Li used it as a black box.

We consider virtual circuit routing protocols with an objective of minimizing energy in a network of components that are speed scalable, and that may be shut down when idle. We assume the standard model for component power: the power consumed by a component with load (speed) $s$ is $\sigma+ s^\alpha$, where $\sigma$ is the static power and the exponent $\alpha>1$. We obtain a very simple $O(\log^\alpha k)$-approximation algorithm for multicommodity routing, where $k$ is the number of demand pairs. This improves upon previous results by several logarithmic factors. The key step in our algorithm is a random sampling technique that we call hallucination, which is reminiscent of the sample-augment framework for buy-at-bulk problems, and sampling in cut-sparsification algorithms. We also consider the online setting of the problem, where demand pairs arrive over time. We show that our offline algorithm naturally extends to the online setting, and obtain a randomized competitive ratio of $\tilde{O}( \log^{3\alpha + 1} k)$, which is the first nontrivial bound. The analysis of this algorithm involves the study of priority multicommodity flows, where edges and demand-pairs have priorities and each demand-pair must route its flow only on edges of lower priority. We establish a polylogarithmic flow-cut gap for these priority flows, which we believe is of independent interest. Finally, we show how our technique can be used to achieve a randomized $( O(\log m), O(\log^2 m))$ bicriteria competitive algorithm for the uniform capacitated network design problem, where $m$ is the number of edges. Here, every edge has a cost $c_e$ and uniform capacity $q$, and the goal is to choose the minimum cost subgraph that can support the given multicommodity demand. This is the first online algorithm for this problem. In fact, our approach also improves prior results in the offline setting by several logarithmic factors.

This paper considers minimizing the $\ell_k$-norms of flow time on a single machine offline using a preemptive scheduler for $k\geq 1$. The objective is ideal for optimizing jobs' overall waiting times while simultaneously being fair to individual jobs. This work gives the first $O(1)$-approximation for the problem, improving upon the previous best $O( \log \log P)$-approximation by Bansal and Pruhs (FOCS 09 and SICOMP 14) where $P$ is the ratio of the maximum job size to the minimum. The main technical ingredient used in this work is a novel combination of quasi-uniform sampling and iterative rounding, which is of interest in its own right.

In this paper we consider one of the most basic scheduling problems where jobs have their respective arrival times and deadlines. The goal is to schedule as many jobs as possible nonpreemptively by their respective deadlines on $m$ identical parallel machines. For the last decade, the best approximation ratio known for the single-machine case ($m = 1$) has been $1-1/e - \epsilon \approx 0.632$ due to Chuzhoy, Ostrovsky, and Rabani [FOCS 2001] and [MOR 2006]. We break this barrier and give an improved 0.644-approximation. For the multiple-machine case, we give an algorithm whose approximation guarantee becomes arbitrarily close to 1 as the number of machines increases. This improves upon the previous best $1 - 1/ (1 + 1/m)^m$ approximation due to Bar-Noy et al. [STOC 1999] and [SICOMP 2009], which converges to 1-1/e as $m$ goes to infinity. Our result for the multiple-machine case extends to the weighted throughput objective where jobs have different weights, and the goal is to schedule jobs with the maximum total weight. Our results show that the 1 - 1/e approximation factor widely observed in various coverage problems is not tight for the nonpreemptive maximum throughput scheduling problem.

Co-flows model a modern scheduling setting that is commonly found in a variety of applications in distributed and cloud computing. In co-flow scheduling, there are $m$ input ports and $m$ output ports. Each co-flow $j \in J$ can be represented by a bipartite graph between the input and output ports, where each edge $(i,o)$ with demand $d_{i,o}^j$ means that $d_{i,o}^j$ units of packets must be delivered from port $i$ to port $o$. To complete co-flow $j$, we must satisfy all of its demands. Due to capacity constraints, a port can only transmit (or receive) one unit of data in unit time. A feasible schedule at each time $t$ must therefore be a bipartite matching. We consider co-flow scheduling and seek to optimize the popular objective of total weighted completion time. Our main result is a $(2+ε)$-approximation for this problem, which is essentially tight, as the problem is hard to approximate within a factor of $(2 - ε)$. This improves upon the previous best known 4-approximation. Further, our result holds even when jobs have release times without any loss in the approximation guarantee. The key idea of our approach is to construct a continuous-time schedule using a configuration linear program and interpret each job's completion time therein as the job's deadline. The continuous-time schedule serves as a witness schedule meeting the discovered deadlines, which allows us to reduce the problem to a deadline-constrained scheduling problem. * This result is flawed; see the first page for the details.

Modern data centers face a key challenge of effectively serving user requests that arrive online. Such requests are inherently multidimensional and characterized by demand vectors over multiple resources such as processor cycles, storage space, and network bandwidth. Typically, different resources require different objectives to be optimized, and $L_r$ norms of loads are among the most popular objectives considered. Furthermore, the server clusters are also often heterogeneous making the scheduling problem more challenging. To address these problems, we consider the online vector scheduling problem in this paper. Introduced by Chekuri and Khanna in 2006, vector scheduling is a generalization of classical load balancing, where every job has a vector load instead of a scalar load. The scalar problem, introduced by Graham in 1966, and its many variants (identical and unrelated machines, makespan and $L_r$ norm optimization, offline and online jobs, etc.) have been extensively studied over the last 50 years. In this paper, we resolve the online complexity of the vector scheduling problem and its important generalizations---for all $L_r$ norms and in both the identical and unrelated machines settings. For an instance with $m$ machines and $d$ dimensions, our main results are: For identical machines, we show that the optimal competitive ratio is $\Theta(\log d / \log \log d)$ by giving an online lower bound and an algorithm with an asymptotically matching competitive ratio. The lower bound is technically challenging, and is obtained via an online lower bound for the minimum monochromatic clique problem using a novel online coloring game and randomized coding scheme. Our techniques also extend to asymptotically tight upper and lower bounds for general $L_r$ norms. For unrelated machines, we show that the optimal competitive ratio is $\Theta(\log m + \log d)$ by giving an online lower bound that matches a previously known upper bound. Unlike identical machines, however, extending these results, particularly the upper bound, to general $L_r$ norms requires new ideas. In particular, we use a carefully constructed potential function that balances the individual $L_r$ objectives with the overall (convexified) min-max objective to guide the online algorithm and track the changes in potential to bound the competitive ratio.

Coflow has recently been introduced to capture communication patterns that are widely observed in the cloud and massively parallel computing. Coflow consists of a number of flows that each represents data communication from one machine to another. A coflow is completed when all of its flows are completed. Due to its elegant abstraction of the complicated communication processes found in various parallel computing platforms, it has received significant attention. In this paper, we consider coflow for the objective of maximizing partial throughput. This objective seeks to measure the progress made for partially completed coflows before their deadline. Partially processed coflows still could be useful when their flows send out useful data that can be used for the next round computation. In our measure, a coflow is processed by a certain fraction when all of its flows are processed by the same fraction or more. We consider a natural class of greedy algorithms, which we call myopic concurrent. The algorithms seek to maximize the marginal increase of the partial throughput objective at each time. We analyze the performance of our algorithm against the optimal scheduler. In fact, our result is more general as a flow could be extended to demand various heterogeneous resources. Our experiment demonstrates our algorithm's superior performance.

In this paper, we consider the problem of assigning jobs online to machines with non-uniform speeds (also called related machines) so to optimize a given norm of the machine loads. A long line of work, starting with the seminal work of Graham in the 1960s, has led to tight competitive ratios for all ℓq norms for two scenarios: the special case of identical machines (uniform machine speeds) and the more general setting of unrelated machines (jobs have arbitrary processing times on machines). For non-uniform machine speeds, however, the only known result was a constant competitive competitive ratio for the makespan (ℓ∞) norm, via the so-called slowest-fit algorithm (Aspnes, Azar, Fiat, Plotkin, and Waarts, JACM ’97). Our first result in this paper is to obtain the first constant-competitive algorithm for scheduling on related machines for any arbitrary ℓq norm.

We introduce and study a general scheduling problem that we term the Polytope Scheduling problem (PSP). In this problem, jobs can have different arrival times and sizes, and the rates assigned by the scheduler to the jobs are subject to arbitrary packing constraints. The PSP framework captures a variety of scheduling problems, including the classical problems of unrelated machines scheduling, broadcast scheduling, and scheduling jobs of different parallelizability. It also captures scheduling constraints arising in diverse modern environments ranging from individual computer architectures to data centers. More concretely, PSP models multidimensional resource requirements and parallelizability, as well as network bandwidth requirements found in data center scheduling. We show a surprising result—there is a single algorithm that is O (1) competitive for all PSP instances when the objective is total completion time, and O (1) competitive for a large sub-class of PSP instances when the objective is total flow time. This algorithm simply uses the well-known Proportional Fairness (PF) algorithm to perform allocations each time instant. Though P F has been extensively studied in the context of maximizing fairness in resource allocation, we present the first analysis in adversarial and general settings for optimizing job latency. Further, P F is non-clairvoyant, meaning that the algorithm doesn’t need to know jobs sizes until their completion. We establish our positive results by making novel connections with Economics, in particular, the notions of market clearing, Gross Substitutes, and Eisenberg-Gale markets. We complement these positive results with a negative result: We show that for the total flow time objective, any non-clairvoyant algorithm for general PSP has a strong lower bound on the competitive ratio unless given a poly-logarithmic speed augmentation. This motivates the need to consider sub-classes of PSP when studying flow time. The sub-class for which we obtain positive results not only captures several well-studied models, such as scheduling with speedup curves and related machine scheduling, but also captures as special cases hitherto unstudied scheduling problems, such as single source flow routing, routing multicast (video-on-demand) trees, and resource allocation with substitute resources.

We consider a setting where selfish agents want to schedule jobs on related machines. The agent submitting a job picks a server that minimizes a linear combination of the server price and the resulting response time for that job on the selected server. The manager's task is to maintain server prices to (approximately) optimize the maximum response time, which is a measure of social good. We show that the existence of a pricing scheme with certain competitiveness is equivalent to the existence of a monotone immediate-dispatch algorithm. Our main result is a monotone immediate-dispatch algorithm that is O(1)-competitive with respect to the maximum response time.

This paper considers the online machine minimization problem, a basic real time scheduling problem. The setting for this problem consists of n jobs that arrive over time, where each job has a deadline by which it must be completed. The goal is to design an online scheduler that feasibly schedules the jobs on a nearly minimal number of machines. An algorithm is c-machine optimal if the algorithm will feasibly schedule a collection of jobs on c ·m machines if there exists a feasible schedule on m machines. For over two decades the best known result was a O(log P)-machine optimal algorithm, where P is the ratio of the maximum to minimum job size. In a recent breakthrough, a O(log m)-machine optimal algorithm was given. In this paper, we exponentially improve on this recent result by giving a O(log log m)-machine optimal algorithm.

In the paper we consider minimizing the ℓk-norms of flow time on a single machine offline using a preemptive scheduler for k ≥ 1. We show the first O( 1)- approximation for the problem, improving upon the previous best O(log log P)-approximation by Bansal and Pruhs (FOCS 09 and SICOMP 14) where P is the ratio of the maximum job size to the minimum. Our main technical ingredient is a novel combination of quasi-uniform sampling and iterative rounding, which is of interest in its own right.

Modern science and engineering is driven by massively large data sets and its advance heavily relies on massively parallel computing platforms such as Spark, MapReduce, and Hadoop. Theoretical models have been proposed to understand the power and limitations of such platforms. Recent study of developed theoretical models has led to the discovery of new algorithms that are fast and efficient in both theory and practice, thereby beginning to unlock their underlying power. Given recent promising results, the area has turned its focus on discovering widely applicable algorithmic techniques for solving problems efficiently.

Modern data centers consist of a large number of heterogeneous resources such as CPU, memory, network bandwidth, etc. The resources are pooled into clusters for various reasons such as scalability, resource consolidation, and privacy. Clusters are often heterogeneous so that they can better serve jobs with different characteristics submitted from clients. Each job benefits differently depending on how much resource is allocated to the job, which in turn translates to how quickly the job gets completed. In this paper, we formulate this setting, which we term Multi-Cluster Polytope Scheduling (MCPS). In MCPS, a set of n jobs arrive over time to be executed on m clusters. Each cluster i is associated with a polytope P_i, which constrains how fast one can process jobs assigned to the cluster. For MCPS, we seek to optimize the popular objective of minimizing average weighted flow time of jobs in the online setting. We give a constant competitive algorithm with small constant resource augmentation for a large class of polytopes, which capture many interesting problems that arise in practice. Further, our algorithm is non-clairvoyant. Our algorithm and analysis combine and generalize techniques developed in the recent results for the classical unrelated machines scheduling and the polytope scheduling problem [10,12,11].

In this paper we consider the classic scheduling problem of minimizing total weighted completion time on unrelated machines when jobs have release times, i.e, R|rij| ΣjwjCjusing the three-field notation. For this problem, a 2-approximation is known based on a novel convex programming (J. ACM 2001 by Skutella). It has been a long standing open problem if one can improve upon this 2-approximation (Open Problem 8 in J. of Sched. 1999 by Schuurman and Woeginger). We answer this question in the affirmative by giving a 1.8786-approximation. We achieve this via a surprisingly simple linear programming, but a novel rounding algorithm and analysis. A key ingredient of our algorithm is the use of random offsets sampled from non-uniform distributions. We also consider the preemptive version of the problem, i.e, R|rij, pmtn|ΣjwjCj. We again use the idea of sampling offsets from non-uniform distributions to give the first better than 2-approximation for this problem. This improvement also requires use of a configuration LP with variables for each job's complete schedules along with more careful analysis. For both non-preemptive and preemptive versions, we break the approximation barrier of 2 for the first time.

We consider the classical problem of constrained queueing (or switched networks): There is a set of N queues to which unit sized packets arrive. The queues are interdependent, so that at any time step, only a subset of the queues can be activated. One packet from each activated queue can be transmitted, and leaves the system. The set of feasible subsets that can be activated, denoted S, is downward closed and is known in advance. The goal is to find a scheduling policy that minimizes average delay (or flow time) of the packets. The constrained queueing problem models several practical settings including packet transmission in wireless networks and scheduling cross-bar switches. In this paper, we study this problem using the the competitive analysis: The packet arrivals can be adversarial and the scheduling policy only uses information about packets currently queued in the system. We present an online algorithm, that for any epsilon > 0, has average flow time at most O(R^2/epsilon^3*OPT+NR) when given (1+epsilon) speed, i.e., the ability to schedule (1+epsilon) packets on average per time step. Here, R is the maximum number of queues that can be simultaneously scheduled, and OPT is the average flow time of the optimal policy. This asymptotic competitive ratio O(R^3/epsilon^3) improves upon the previous O(N/epsilon^2) which was obtained in the context of multi-dimensional scheduling [Im/Kulkarni/Munagala, FOCS 2015]. In the full general model where N can be exponentially larger than R, this is an exponential improvement. The algorithm presented in this paper is based on Makespan estimates which is very different from that in [Im/Kulkarni/Munagala, FOCS 2015], a variation of the Max-Weight algorithm. Further, our policy is myopic, meaning that scheduling decisions at any step are based only on the current composition of the queues. We finally show that speed augmentation is necessary to achieve any bounded competitive ratio.

We consider the problem of scheduling jobs with varying demands on multiple servers. Each server has a certain computing capacity and can schedule multiple jobs simultaneously as long as the jobs' total demand does not exceed the server's capacity. This scenario arises commonly in virtualization, cloud computing, and MapReduce (or Hadoop). We study this problem with the requirement that jobs must be scheduled non-preemptively, meaning that every job must be completed without interruption once it gets started. Often, preemption is out of choice since preempting a job can be prohibitively costly or is not allowed due to system constraints. We focus on the popular objective of minimizing total completion time of jobs. This problem is NP hard hence we study heuristics with provable approximation guarantees. Succinctly, the interaction between two orthogonal quantities, jobs demands and sizes makes the scheduling decision significantly more challenging. In this paper we propose novel algorithms for scheduling jobs with non-uniform demands on multiple homogeneous servers without preemption. We first observe that the Smallest Volume First (SVF) algorithm that favors jobs with smaller volumes could perform very poorly in general. However, we show that SVF yields a nearly optimal schedule when the system is overloaded and jobs have demands considerably smaller than servers' capacities. This result supports the intuition that SVF should work well unless some jobs with high demands occupy the servers for long, blocking other jobs. Building on this intuition and using reduction to geometric packing problems, we develop algorithms that are constant approximation for all instances for the first time. Prior to our work, there was no theoretical study on this problem even for the single server case.

Scheduling jobs on multiple machines has numerous applications and has been a central topic of research in the scheduling literature. Recently, much progress has been made particularly in online scheduling with the development of powerful analysis tools. In this line of wok a centralized scheduler typically dispatches jobs to machines to exploit the given resources the best to achieve the best system performance which is measured by a certain global scheduling objective. While this approach has been very successful in attacking scheduling problems of growing complexity, the underlying assumption that jobs follow a centralized scheduler may not be realistic in certain scheduling settings. In this paper we initiate the study of online scheduling for selfish jobs in the presence of multiple machines. Selfish behavior of jobs is a common aspect observed in the absence of a centralized scheduler. We explore this question in the unrelated machines setting, arguably one of the most general multiple machine models. In this model each job can have a completely different processing time on each machine. Motivated by several practical scenarios, we assume that when a job arrives it chooses the machine that completes the job the earliest i.e. minimizes the flow time of the job. The goal is to design a local scheduling algorithm on each machine with the goal of minimizing the total (weighted) flow time. We show that the algorithm Smoothed Latest Arrival Processor Sharing, which was introduced in a recent work by Im et al. [27,28], yields an O(1 / ε2)-competitive schedule when given (1 + ε) speed. We also extend our result to minimize total flow-time plus energy consumed. To show this result we establish several interesting properties of the algorithm which could be of potential use for other scheduling problems.

In this paper we consider the power of migration in heterogeneous machines settings and general profit scheduling. We begin by showing that on related machines or on related machines with restricted assignment that any migratory algorithm can be simulated by a non-migratory algorithm given 1+ε speed augmentation and O(1/ε) and O(1/ε2) machine augmentation, respectively, for any 0 < ε ≤ 1. Similar results were only known in the case of identical machines and our results effectively show that migration does not give too much additional power to an algorithm, even in heterogeneous environments. Our results are constructive and can be computed efficiently in the offline setting. We complement our result by showing that there exists migratory schedules on related machines which require Ω(1/ε) machine augmentation with (1+ε)-speed to be simulated by any non-migratory scheduler for any 0 < ε ≤ 1/2, showing that machine augmentation without speed augmentation is insufficient for a non-migratory scheduler to simulate a migratory scheduler. We then use these results to study general profit scheduling where a set of n jobs arrive over time online and every job i has a function gi(t) specifying the profit of completing job i at time t. The goal of the schedule is to maximize the total profit obtained. We give a (1+ε)-speed O(1/ε2)-competitive algorithm in the unrelated machines setting for any ε >0 when compared against a non-migratory adversary. Previous results were only known in the identical machines setting. As an example of the usefulness of the previous results on migration, they with the results on genial profit scheduling give a (1+ε)-speed O(1/ε4)-competitive algorithm for general profit scheduling when comparing against a migratory algorithm on related machines with restricted assignment for any ε >0.

Modern processors typically allow dynamic speed-scaling offering an effective trade-off between high throughput and energy efficiency. In a classical model, a processor/machine runs at speed s when consuming power sα where α >1 is a constant. Yao et al. [FOCS 1995] studied the problem of completing all jobs before their deadlines on a single machine with the minimum energy in their seminal work and gave a nice polynomial time algorithm. The influential work has been extended to various settings. In particular, the problem has been extensively studied in the presence of multiple machines as multi-core processors have become dominant computing units. However, when jobs must be scheduled non-preemptively, our understanding of the problem remains fairly unsatisfactory. Often, preempting a job is prohibited since it could be very costly. Previously, a O((wmax wmin)α)-approximation was known for the non-preemptive setting where wmax and wmin denote the maximum and minimum job sizes, respectively. Even when there is only one machine, the best known approximation factor had a dependency on α. In this paper, for any fixed α >1 and ε >0, we give the first (1+ε)-approximation for this problem on multiple machines which runs in nO(polylog (n)) time where n is the number of jobs to be scheduled.

We study the Minimum Latency Submodular Cover (MLSC) problem, which consists of a metric ( V , d ) with source r ∈ V and m monotone submodular functions f 1 , f 2 , …, f m : 2 V → [0, 1]. The goal is to find a path originating at r that minimizes the total “cover time” of all functions. This generalizes well-studied problems, such as Submodular Ranking [Azar and Gamzu 2011] and the Group Steiner Tree [Garg et al. 2000]. We give a polynomial time O (log 1/ϵ ċ log 2+δ |V|)-approximation algorithm for MLSC, where ϵ > 0 is the smallest non-zero marginal increase of any { f i } m i = 1 and δ > 0 is any constant. We also consider the Latency Covering Steiner Tree (LCST) problem, which is the special case of MLSC where the f i s are multi-coverage functions. This is a common generalization of the Latency Group Steiner Tree [Gupta et al. 2010; Chakrabarty and Swamy 2011] and Generalized Min-sum Set Cover [Azar et al. 2009; Bansal et al. 2010] problems. We obtain an O (log 2 | V |)-approximation algorithm for LCST. Finally, we study a natural stochastic extension of the Submodular Ranking problem and obtain an adaptive algorithm with an O (log 1/ϵ)-approximation ratio, which is best possible. This result also generalizes some previously studied stochastic optimization problems, such as Stochastic Set Cover [Goemans and Vondrák 2006] and Shared Filter Evaluation [Munagala et al. 2007; Liu et al. 2008].

Modern data centers face a key challenge of effectively serving user requests that arrive online. Such requests are inherently multi-dimensional and characterized by demand vectors over multiple resources such as processor cycles, storage space, and network bandwidth. Typically, different resources require different objectives to be optimized, and Lrnorms of loads are among the most popular objectives considered. Furthermore, the server clusters are also often heterogeneous making the scheduling problem more challenging. To address these problems, we consider the online vector scheduling problem in this paper. Introduced by Chekuri and Khanna (SIAM J. of Comp. 2006), vector scheduling is a generalization of classical load balancing, where every job has a vector load instead of a scalar load. The scalar problem, introduced by Graham in 1966, and its many variants (identical and unrelated machines, makespan and Lr-norm optimization, offline and online jobs, etc.) have been extensively studied over the last 50 years. In this paper, we resolve the online complexity of the vector scheduling problem and its important generalizations - for all Lrnorms and in both the identical and unrelated machines settings. Our main results are: · For identical machines, we show that the optimal competitive ratio is Θ(log d/ log log d) by giving an online lower bound and an algorithm with an asymptotically matching competitive ratio. The lower bound is technically challenging, and is obtained via an online lower bound for the minimum mono-chromatic clique problem using a novel online coloring game and randomized coding scheme. Our techniques also extend to asymptotically tight upper and lower bounds for general Lrnorms. · For unrelated machines, we show that the optimal competitive ratio is Θ(log m + log d) by giving an online lower bound that matches a previously known upper bound. Unlike identical machines, however, extending these results, particularly the upper bound, to general Lrnorms requires new ideas. In particular, we use a carefully constructed potential function that balances the individual Lrobjectives with the overall (convexified) min-max objective to guide the online algorithm and track the changes in potential to bound the competitive ratio.

Many scheduling problems can be viewed as allocating rates to jobs, subject to convex packing constraints on the rates. In this paper, we consider the problem of rate allocation when jobs of unknown size arrive online (non-clairvoyant setting), with the goal of minimizing weighted delay or flow time. Though this problem has strong lower bounds on competitive ratio in its full generality, we show positive results for natural and fairly broad sub-classes. More specifically, the subclasses we consider not only generalize several well-studied models such as scheduling with speedup curves and related machine scheduling, but also capture as special cases hitherto unstudied scheduling problems such as routing multi-commodity flows, routing multicast (video-on-demand) trees, and multi-dimensional resource allocation. We establish several first positive results by making connections with two disparate disciplines: Economics and Queueing theory. First, we view the instantaneous allocation of rates as a resource allocation problem. We analyze the natural proportional fairness algorithm from economics. To do this, we extend results from market clearing literature, particularly the Eisenberg-Gale markets and the notions of Walrasian equilibria and Gross Substitutes. This yields the first constant competitive algorithm with constant speed augmentation for single-sink flow routing, routing multicast trees, and multidimensional resource allocation with substitutes resources. Next, we consider the general scheduling problem with packing constraints on rates, but with the restriction that the number of different job types is fixed. We model this problem as a non-stochastic queueing problem. We generalize a natural algorithm from queueing literature and analyze it by extending queueing theoretic ideas. We show that the competitive ratio, for any constant speed, depends polynomially only on the number of job types. Further, such a dependence on the number of job types is unavoidable for non-clairvoyant algorithms. This yields the first algorithm for scheduling multicommodity flows whose competitive ratio depends polynomially on the size of the underlying graph, and not on the number of jobs.

In this paper, we consider a variety of scheduling problems where n jobs with release times are to be scheduled non-preemptively on a set of m identical machines. The problems considered are machine minimization, (weighted) throughput maximization and min-sum objectives such as (weighted) flow time and (weighted) tardiness. We develop a novel quasi-polynomial time dynamic programming framework that gives O(l)-speed O(l)-approximation algorithms for the offline versions of machine minimization and min-sum problems. For the weighted throughput problem, the framework gives a (1 + ε)-speed (1 – ε)-approximation algorithm. The generic DP is based on improving a naïve exponential time DP by developing a sketching scheme that compactly and accurately approximates parameters used in the DP states. We show that the loss of information due to the sketching scheme can be offset with limited resource augmentation. This framework is powerful and flexible, allowing us to apply it to this wide range of scheduling objectives and settings. We also provide new insight into the relative power of speed augmentation versus machine augmentation for non-preemptive scheduling problems; specifically, we give new evidence for the power and importance of extra speed for some non-preemptive scheduling problems. This novel DP framework leads to many new algorithms with improved results that solve many open problems, albeit with quasi-polynomial running times. We highlight our results as follows. For the problems with min-sum objectives, we give the first O(l)-speed O(l)-approximation algorithms for the multiple-machine setting. Even for the single machine case, we reduce both the resource augmentation required and the approximation ratios. In particular, our approximation ratios are either 1 or 1 + ε. Most of our algorithms use speed 1 + e or 2 + ε. We also resolve an open question (albeit with a quasi-polynomial time algorithm) of whether less than 2-speed could be used to achieve an O(1)-approximation for flow time. New techniques are needed to address this open question since it was proven that previous techniques are insufficient. We answer this open question by giving an algorithm that achieves a (1 + ε)-speed 1-approximation for flow time and (1 + ε)-speed (1 + ε)-approximation for weighted flow time. For the machine minimization problem, we give the first result using constant resource augmentation by showing a (1 + ε)-speed 2-approximation, and the first result only using speed augmentation and no additional machines by showing a (2 + ε)-speed 1-approximation. We complement our positive results for machine minimization by considering the discrete variant of the problem and show that no algorithm can use speed augmentation less than 2log1–εand achieve approximation less than O(log log n) for any constant ε > 0 unless NP admits quasi-polynomial time optimal algorithms. Thus, our results show a stark contrast between the two settings. In one, constant speed augmentation is sufficient whereas in the other, speed augmentation is essentially not effective.

We revisit the pull-based broadcast scheduling model. In this model, there are n unit-sized pages of information available at the server. Clients send their requests to the server over time asking for specific pages. The server can transmit only one page at each time. When the server transmits a page, all outstanding requests for the page are simultaneously satisfied, and this is what distinguishes broadcast scheduling from the standard scheduling setting where each job must be processed separately by the server. Broadcast scheduling has received a considerable amount of attention due to the algorithmic challenges that it gives in addition to its applications in multicast systems and wireless and LAN networks. In this paper, we give the following new approximation results for two popular objectives: For the objective of minimizing the maximum flow time, we give the first PTAS. Previously, it was known that the algorithm First-In-First-Out (FIFO) is a 2-approximation, and it is tight [14, 16]. It has been suggested as an open problem to obtain a better approximation [14, 4, 25, 31]. For the objective of maximizing the throughput, we give a 0.7759-approximation which improves upon the previous best known 0.75-approximation [23]. Our key techniques for these improvements are novel variants of α-point rounding that can effectively reduce congestion in schedule which is often the main hurdle in designing scheduling algorithms based on linear programming. We believe that our new rounding schemes could be of potential use for other scheduling problems.

Fairness is an important criterion considered in scheduling together with overall job latency. Round Robin is a popular scheduling policy that distributes resources to jobs equally at any point in time guaranteeing instantaneous fairness of jobs. In this paper we give the first analysis of Round Robin for the L_2-norm of flow time and show that it is O(1)-speed O(1)-competitive on multiple machines. The L_2-norm is a popular scheduling objective that makes a natural balance between temporal fairness and jobs latency. Prior to our work, Round Robin has not been analyzed for the L_2-norm even in the single machine setting. Our result establishes that Round Robin is fair not only instantaneously but also temporarily.

In this paper we introduce a new network scheduling model. Here jobs need to be sent via routers on a tree to machines to be scheduled, and the communication is constrained by network bandwidth. The scheduler coordinates network communication and job machine scheduling. This type of scheduler is highly desirable in practice; yet few works have considered combing networking with job processing. We consider the popular objective of total flow time in the online setting. We give a (1+ε)-speed O(1/ε7)-competitive algorithm when all routers are identical and all machines are identical for any fixed ε >0. Then we go on to show a (2+ε)-speed O(1/ε7)-competitive algorithm when the routers are identical and the machines are unrelated. To show these results we introduce an interesting combination of potential function and dual fitting techniques as well as a reduction of general tree scheduling to a special case of trees.

In this paper we introduce a new network scheduling model. Here jobs need to be sent via routers on a tree to machines to be scheduled, and the communication is constrained by network bandwidth. The scheduler coordinates network communication and job machine scheduling. This type of scheduler is highly desirable in practice; yet few works have considered combing networking with job processing. We consider the popular objective of total flow time in the online setting. We give a (1+ε)-speed O(1/ε7)-competitive algorithm when all routers are identical and all machines are identical for any fixed eps >0. Then we go on to show a (2+ε)-speed O(1/ε7)-competitive algorithm when the routers are identical and the machines are unrelated. To show these results we introduce an interesting combination of potential function and dual fitting techniques as well as a reduction of general tree scheduling to a special case of trees.

We revisit the classical stochastic scheduling problem of nonpreemptively scheduling n jobs so as to minimize total completion time on m identical machines, P \mid \mid \mathbb{E} \sum C_j in the standard 3-field scheduling notation. Previously it was only known how to obtain reasonable approximation if jobs sizes have low variability. However, distributions commonly arising in practice have high variability, and the upper bounds on the approximation ratio for the previous algorithms for such distributions can be even inverse-polynomial in the maximum possible job size. We start by showing that the natural list scheduling algorithm Shortest Expected Processing Time (SEPT) has a bad approximation ratio for high variability jobs. We observe that a simple randomized rounding of a natural linear programming relaxation is a (1+\epsilon)-machine O(1)-approximation assuming the number of machines is at least logarithmic in the number of jobs. Turning to the case of a modest number of machines, we develop a list scheduling algorithm that is O(\log^2 n + m \log n)-approximate. Our results together imply a (1+\epsilon)-machine O(\log^2 n )-approximation for an arbitrary number of machines. Intuitively our list scheduling algorithm finds an ordering that not only takes the expected size of a job into account, but also takes into account the probability that job will be big.

We consider the classical problem of minimizing the total weighted flow-time for unrelated machines in the online non-clairvoyant setting. In this problem, a set of jobs J arrive over time to be scheduled on a set of M machines. Each job J has processing length pj, weight wj, and is processed at a rate of lij when scheduled on machine i. The online scheduler knows the values of wj and lij upon arrival of the job, but is not aware of the quantity pj. We present the first online algorithm that is scalable ((1+ε)-speed O(1/2)-competitive for any constant ε > 0) for the total weighted flow-time objective. No non-trivial results were known for this setting, except for the most basic case of identical machines. Our result resolves a major open problem in online scheduling theory. Moreover, we also show that no job needs more than a logarithmic number of migrations. We further extend our result and give a scalable algorithm for the objective of minimizing total weighted flow-time plus energy cost for the case of unrelated machines. In this problem, each machine can be sped up by a factor of f-1i(P) when consuming power P, where fi is an arbitrary strictly convex power function. In particular, we get an O(γ2)-competitive algorithm when all power functions are of form sγ. These are the first non-trivial non-clairvoyant results in any setting with heterogeneous machines. The key algorithmic idea is to let jobs migrate selfishly until they converge to an equilibrium. Towards this end, we define a game where each job's utility which is closely tied to the instantaneous increase in the objective the job is responsible for, and each machine declares a policy that assigns priorities to jobs based on when they migrate to it, and the execution speeds. This has a spirit similar to coordination mechanisms that attempt to achieve near optimum welfare in the presence of selfish agents (jobs). To the best our knowledge, this is the first work that demonstrates the usefulness of ideas from coordination mechanisms and Nash equilibria for designing and analyzing online algorithms.

We study the price of anarchy of coordination mechanisms for a scheduling problem where each job j has a weight wj, processing time pij, assignment cost hij, and communication delay (or release date) rij, on machine i. Each machine is free to declare its own scheduling policy. Each job is a selfish agent and selects a machine that minimizes its own disutility, which is equal to its weighted completion time plus its assignment cost. The goal is to minimize the total disutility incurred by all the jobs. Our model is general enough to capture scheduling jobs in a distributed environment with heterogeneous machines (or data centers) that are situated across different locations.

We consider virtual circuit routing protocols, with an objective of minimizing energy, in a network of components that are speed scalable, and that may be shutdown when idle. We assume that the speed s of a link is proportional to its load, and assume the standard model for component power, namely that the power is some constant static power σ plus sα, where typically α ∊ [1.1,3]. We give a polynomial-time offline algorithm for multicommodity routing, that has approximation ratio O(loga k), where k is the number of demand pairs. This is obtained as a combination of three natural combinatorial algorithms. The key step of the algorithm design is a random sampling technique that we call hallucination, which is reminiscent of the Sample-Augment framework for solving Buy-at-Bulk type problems, and sampling in cut-sparsification algorithms. The analysis of the approximation ratio is then a direct consequence of the flow-cut gap for multicommodity flow. The algorithm extends rather naturally to an online algorithm, which we show has competitive ratio Õ(log3a+1 k). The analysis of the online algorithm introduces a natural “priority” multicommodity flow problem, and bounds the priority multicommodity flow-cut gap-this might also be of independent interest. We also explain how our hallucination technique can be used to achieve an (O(log km), O(logkm)) bicriteria approximation result for the problem of buying a minimum cost collection of unit-capacitated edges to support a concurrent multicommodity flow, where m is the number of links in the network.

In this paper we consider the buffer reordering management problem. In this model there are n elements that arrive over time with different colors. There is a buffer that can store up to k elements and when the buffer becomes full an element must be output. If an element is output that has a color different from the previous element, a cost depending on the color must be paid. This cost could be uniform or non-uniform over colors; these are called unweighted and weighted cases, respectively. The goal is to reorder elements within the buffer before outputting them to minimize the total cost incurred. There has been a search over the last decade to resolve the complexity of this problem online and offline. Very recently, there has been substantial progress for the unweighted case – an O(1)-approximation algorithm and an O(log log k)-competitive randomized algorithm were given [6, 7]. These results resolve the complexity of the unweighted buffer problem, up to constant factors, since the problem is NP-Hard and there is a matching lower bound on the competitive ratio. However, the progress for the weighted case has not been as satisfactory as for the unweighted case. Our main result is a randomized O(loglogkγ)-approximation for the weighted case, which gives an exponential improvement over the previously best known result of O(y/logk) which assumed γ = poly(k). Here γ is the ratio of the maximum to minimum weight. We also revisit the unweighted case and give an improved randomized 66.0823-approximation which improves (modestly) upon the approximation guarantee given in [6]. The algorithm and analysis we use for the unweighted case was done independently of [6]. We believe that our new interpretation of the problem and our analysis of an underlying random process could be of potential use in other settings.

We introduce a scheduling algorithm Intermediate-SRPT, and show that it is O(log P)-competitive with respect to average waiting time when scheduling jobs whose parallelizability is intermediate between being fully parallelizable and sequential. Here the parameter P denotes the ratio between the maximum job size to the minimum. We also show a general matching lower bound on the competitive ratio. Our analysis builds on an interesting combination of potential function and local competitiveness arguments.

We introduce and study a general scheduling problem that we term the Packing Scheduling problem (PSP). In this problem, jobs can have different arrival times and sizes; a scheduler can process job j at rate xj, subject to arbitrary packing constraints over the set of rates (x) of the outstanding jobs. The PSP framework captures a variety of scheduling problems, including the classical problems of unrelated machines scheduling, broadcast scheduling, and scheduling jobs of different parallelizability. It also captures scheduling constraints arising in diverse modern environments ranging from individual computer architectures to data centers. More concretely, PSP models multidimensional resource requirements and parallelizability, as well as network bandwidth requirements found in data center scheduling.

We consider a general online scheduling problem where the goal is to minimize $\sum_j w_j g(F_j)$, where $w_j$ is the weight/importance of job $J_j$, $F_j$ is the flow time of the job in the schedule, and $g$ is an arbitrary nondecreasing cost function. Numerous natural scheduling objectives are special cases of this general framework. We show that the scheduling algorithm Highest Density First (HDF) is $(2+\epsilon)$-speed $O(1)$-competitive for all cost functions $g$ simultaneously. We give lower bounds that show that the HDF algorithm and this analysis are essentially optimal. Finally, we show that scalable algorithms are achievable in some special cases.

Integrated Modular Avionics (IMA) architecture has been widely adopted by the avionics industry due to its strong temporal and spatial isolation capability for safety-critical real-time systems. The fundamental challenge to integrating an existing set of single-core IMA partitions into a multi-core system is to ensure that the isolation of the partitions will be maintained without incurring huge redevelopment and recertification costs. To address this challenge, we developed an optimized partition scheduling algorithm which considers exclusive regions to achieve the synchronization between partitions across cores. We show that the problem of finding the optimal partition schedule is NP-complete and present a Constraint Programming formulation. In addition, we relax this problem to find the minimum number of cores needed to schedule a given set of partitions and propose an approximation algorithm which is guaranteed to find a feasible schedule of partitions if there exists a feasible schedule of exclusive regions.

In this paper, we consider scheduling parallelizable jobs in the non-clairvoyant speed scaling setting to minimize the objective of weighted flow time plus energy. Previously, strong lower bounds were shown on this model in the unweighted setting even when the algorithm is given a constant amount of resource augmentation over the optimal solution. However, these lower bounds were given only for certain families of algorithms that do not recognize the parallelizability of alive jobs. In this work, we circumvent previous lower bounds shown and give a scalable algorithm under the natural assumption that the algorithm can know the current parallelizability of a job. When a general power function is considered, this is also the first algorithm that has a constant competitive ratio for the problem using any amount of resource augmentation.

Batch scheduling gives a powerful way of increasing the throughput by aggregating multiple homogeneous jobs. It has applications in large scale manufacturing as well as in server scheduling. In batch scheduling, when explained in the setting of server scheduling, the server can process requests of the same type up to a certain number simultaneously. Batch scheduling can be seen as capacitated broadcast scheduling, a popular model considered in scheduling theory. In this paper, we consider an online batch scheduling model. For this model we address flow time objectives for the first time and give positive results for average flow time, the k-norms of flow time and maximum flow time. For average flow time and the k-norms of flow time we show algorithms that are O(1)-competitive with a small constant amount of resource augmentation. For maximum flow time we show a 2-competitive algorithm and this is the best possible competitive ratio for any online algorithm.

We consider preemptive online scheduling algorithms to minimize the total weighted/unweighted flow time plus energy for speed-scalable heterogeneous multiprocessors. We show that the well-known priority scheduling algorithms Highest Density First, Weighted Shortest Elapsed Time First, and Weighted Late Arrival Processor Sharing, are not O(1)-speed O(1)-competitive for the objective of weighted flow even in the special case of fixed variable speed processors (aka the related machines setting). This illustrates that scheduling heterogeneous multiprocessors is a different, and algorithmically more challenging problem, than scheduling homogeneous multiprocessors. We then show that a variation of the non-clairvoyant algorithm Late Arrival Processor Sharing coupled with a non-obvious speed scaling algorithm is scalable for the objective of unweighted flow plus energy on speed-scalable multiprocessors. This is the first provably scalable non-clairvoyant algorithm on heterogeneous multiprocessors, even in the related machines setting, for the objective of total (unweighted) flow time.

We consider a general online scheduling problem on a single machine with the objective of minimizing Σj wjg(Fj), where wj is the weight/importance of job Jj, Fj is the flow time of the job in the schedule, and g is an arbitrary non-decreasing cost function. Numerous natural scheduling objectives are special cases of this general objective. We show that the scheduling algorithm Highest Density First (HDF) is (2+∊)-speed O(1)-competitive for all cost functions g simultaneously. We give lower bounds that show the HDF algorithm and this analysis are essentially optimal. Finally, we show scalable algorithms are achievable in some special cases.

In the (non-preemptive) Generalized Min Sum Set Cover Problem, we are given n ground elements and a collection of sets S = {S_1, S_2, ..., S_m} where each set S_i in 2^{[n]} has a positive requirement k(S_i) that has to be fulfilled. We would like to order all elements to minimize the total (weighted) cover time of all sets. The cover time of a set S_i is defined as the first index j in the ordering such that the first j elements in the ordering contain k(S_i) elements in S_i. This problem was introduced by [Azar, Gamzu and Yin, 2009] with interesting motivations in web page ranking and broadcast scheduling. For this problem, constant approximations are known [Bansal, Gupta and Krishnaswamy, 2010][Skutella and Williamson, 2011]. We study the version where preemption is allowed. The difference is that elements can be fractionally scheduled and a set S is covered in the moment when k(S) amount of elements in S are scheduled. We give a 2-approximation for this preemptive problem. Our linear programming and analysis are completely different from [Bansal, Gupta and Krishnaswamy, 2010][Skutella and Williamson, 2011]. We also show that any preemptive solution can be transformed into a non-preemptive one by losing a factor of 6.2 in the objective function. As a byproduct, we obtain an improved 12.4-approximation for the non-preemptive problem.

In this article, the online pull-based broadcast model is considered. In this model, there are n pages of data stored at a server and requests arrive for pages online. When the server broadcasts page p , all outstanding requests for the same page p are simultaneously satisfied. We consider the problem of minimizing average (total) flow time online where all pages are unit-sized. For this problem, there has been a decade-long search for an online algorithm which is scalable, that is, (1 + ϵ)-speed O (1)-competitive for any fixed ϵ > 0. In this article, we give the first analysis of an online scalable algorithm.

Pattern mining is a fundamental problem that has a wide range of applications. In this paper, we study the problem of finding a minimum set of signature patterns that explain all data. In the problem, we are given objects where each object has an item set and a label. A pattern is called a signature pattern if all objects with the pattern have the same label. This problem has many interesting applications such as assertion mining in hardware design and identifying failure causes from various log data. We show that the previous pattern mining methods are not suitable for mining signature patterns and identify the problems. Then we propose a novel pattern enumeration method which we call Pattern Shrink. Our method is strongly coupled with another novel method that is very similar to finding a local optimum with a negligible loss in performance. Our proposed methods show a speedup of more than ten times over the previous methods. Our methods are flexible enough to be extended to mining high confidence patterns, instead of signature patterns.

Clustering problems have numerous applications and are becoming more challenging as the size of the data increases. In this paper, we consider designing clustering algorithms that can be used in MapReduce, the most popular programming environment for processing large datasets. We focus on the practical and popular clustering problems, k-center and k-median. We develop fast clustering algorithms with constant factor approximation guarantees. From a theoretical perspective, we give the first analysis that shows several clustering algorithms are in MRC0, a theoretical MapReduce class introduced by Karloff et al. [26]. Our algorithms use sampling to decrease the data size and they run a time consuming clustering algorithm such as local search or Lloyd's algorithm on the resulting data set. Our algorithms have sufficient flexibility to be used in practice since they run in a constant number of MapReduce rounds. We complement these results by performing experiments using our algorithms. We compare the empirical performance of our algorithms to several sequential and parallel algorithms for the k-median problem. The experiments show that our algorithms' solutions are similar to or better than the other algorithms' solutions. Furthermore, on data sets that are sufficiently large, our algorithms are faster than the other parallel algorithms that we tested.

We address the scheduling model of arbitrary speed-up curves and the broadcast scheduling model. The former occurs when jobs are scheduled in a multi-core system or on a cloud of machines. Here jobs can be sped up when given more processors or machines. However, the parallelizability of the jobs may vary and the algorithm is required to be oblivious of the parallelizability of a job. The latter model is natural in wireless and LAN networks where requests (or jobs) can be simultaneously satisfied together. Both settings are similar in that two schedules can do different amounts of work to satisfy all the jobs. We focus on optimizing the ℓk- norms of flow time. Recently, Gupta et al. [24] gave a (k + ε)-speed O(1)-competitive algorithm for the ℓk norms of flow time in both scheduling settings for fixed k. Inspired by this work, we give the first analysis of a scalable algorithm, i.e. (1 + ε)-speed O(1)-competitive, for all ℓk-norms of flow time in both settings for fixed k and 0 < ε ≤ 1. Both problems have a strong lower bound without resource augmentation, so this is the best result that can be shown in the worst case setting up to a constant factor in the competitive ratio.

We consider the problem of scheduling jobs that arrive online in the unrelated machine model to minimize ℓk norms of weighted flowtime. In the unrelated setting, the processing time and weight of a job depends on the machine it is assigned to, and it is perhaps the most general machine model considered in scheduling literature. Chadha et al. [10] obtained a recent breakthrough result in obtaining the first non-trivial algorithm for minimizing weighted flowtime (that is, the ℓ1 norm) in this very general setting via a novel potential function based analysis. They described a simple algorithm and showed that for any ε > 0 it is (1 + ε)-speed O(1/ε2)-competitive (a scalable algorithm). In this paper we give the first non-trivial and scalable algorithm for minimizing ℓk norms of weighted flowtime in the unrelated machine model; for any ε > 0, the algorithm is O(k/ε2+2/k)-competitive. The algorithm is immediate-dispatch and non-migratory. Our result is of both practical and theoretical interest. Scheduling to minimize ℓk norms of flowtime for some small k > 1 has been shown to balance total response time and fairness, which is desirable in practice. On the theoretical side, ℓk norms for k > 1 pose substantial technical hurdles when compared to when k = 1 even for the single machine case. Our work develops a novel potential function as well as several tools that can be used to lower bound the optimal solution.

The classical secretary problem studies the problem of hiring the best secretary from among the secretaries who arrive in random order by making immediate and irrevocable decisions. After the interesting connection to online mechanism design was found [19, 20], the random order input assumption has been studied for a variety of problems. Babaioff et al. [4] formalized a general version of the secretary problem, namely the matroid secretary problem. In the problem, a secretary corresponds to an element in the universe U. The goal is to select the maximum weight independent set. They conjectured that the matroid secretary problem, for any matroid, allows a constant competitive algorithm. The conjecture remains open. Some constant approximation algorithms are currently known for some special cases of matroids. Another interesting type of secretary problem was studied where elements have non-uniform sizes, as is the case in the knapsack secretary problem [3, 6]. In this paper, we consider two interesting secretary problems. One is when the matroid is a laminar matroid, which generalizes uniform / partition / truncated partition matroids. For the laminar matroid secretary problem, using a novel replacement rule which we call “kick next,” we give the first constant-competitive algorithm. The other is the interval scheduling secretary problem, which generalizes the knapsack secretary problem. In this problem, each job Ji arrives with interval Ii, processing time pi and weight wi. If Ji is accepted, then it must be scheduled during Ii, not necessarily continuously. The goal is to accept the jobs of the maximum total weight which are schedulable. We give a simple O(log D)-competitive algorithm and a nearly matching lower bound on the competitive ratio of any randomized algorithm, where D is the maximum interval length of any job.

In this paper the online pull-based broadcast model is considered. In this model, there are n pages of data stored at a server and requests arrive for pages online. When the server broadcasts page p, all outstanding requests for the same page p are simultaneously satisfied. We consider the problem of minimizing average (total) flow time online where all pages are unit-sized. For this problem, there has been a decade-long search for an online algorithm which is scalable, i.e. (1 + ε)-speed O(1)-competitive for any fixed ε > 0. In this paper, we give the first analysis of an online scalable algorithm.

We give a (2+ε)-speed O(1)-competitive algorithm for scheduling jobs with arbitrary speed-up curves for the l2 norm of flow. We give a similar result for the broadcast setting with varying page sizes.