This paper studies the online scheduling problem of minimizing total flow time for $n$ jobs on $m$ identical machines. A classical $Ω(n)$ lower bound shows that no deterministic single-machine algorithm can beat the trivial greedy, even when $n$ is known in advance. However, this barrier is specific to deterministic algorithms on a single machine, leaving open what randomization, multiple machines, or the kill-and-restart capability can achieve.
We give a nearly complete answer. For randomized non-preemptive algorithms, we establish a tight $Θ(\sqrt{n/m})$ competitive ratio, which also improves the best offline approximation to $O(\sqrt{n/m})$. For deterministic non-preemptive algorithms on multiple machines, we prove an $O(n/m^2 + \sqrt{n/m}\log m)$ upper bound and an $Ω(n/m^2 + \sqrt{n/m})$ lower bound. In the kill-and-restart model, we reveal a sharp transition for deterministic algorithms: $Ω(n/\log n)$ for $m = 1$ versus $Θ(\sqrt{n/m})$ for $m \ge 2$; the latter matches the optimal randomized ratio, and we further show that randomization provides no additional power in this model.
We also investigate the setting where $n$ is unknown. We prove that no randomized non-preemptive algorithm achieves $o(n)$ on one machine or $o(n/m^2 + \sqrt{n/m})$ on $m$ machines. In contrast, our kill-and-restart algorithm achieves $O(n^α/\sqrt{m})$ for $m \ge 2$, where $α= (\sqrt{5}-1)/2$, breaking the trivial bound without knowledge of $n$.
We study a scheduling problem arising in demand response management in smart grid. Consumers send in power requests with a flexible feasible time interval during which their requests can be served. The grid controller, upon receiving power requests, schedules each request within the specified interval. The electricity cost is measured by a convex function of the load in each timeslot. The objective is to schedule all requests with the minimum total electricity cost. Previous work has studied cases where jobs have unit power requirement and unit duration. We extend the study to arbitrary power …
We consider the online problem of minimizing weighted flow-time on unrelated machines. Although much is known about this problem in the resource-augmentation setting, these results assume that jobs can be preempted. We give the first constant-competitive algorithm for the non-preemptive setting in the rejection model. In this rejection model, we are allowed to reject an $\varepsilon$-fraction of the total weight of jobs, and compare the resulting flow-time to that of the offline optimum which is required to schedule all jobs. This is arguably the weakest assumption in which such a result is kn…
We consider the classical problem of scheduling $n$ jobs with release dates on both single and identical parallel machines. We measure the quality of service provided to each job by its stretch, which is defined as the ratio of its response time to processing time. Our objective is to schedule these jobs non-preemptively so as to minimize sum stretch. So far, there have been very few results for sum stretch minimization especially for the non-preemptive case. For the preemptive version, the Shortest remaining processing time (SRPT) algorithm is known to give $2$-competitive for sum stretch on …
We consider the online resource minimization problem in which jobs with hard deadlines arrive online over time at their release dates. The task is to determine a feasible schedule on a minimum number of machines. We rigorously study this problem and derive various algorithms with small constant competitive ratios for interesting restricted problem variants. As the most important special case, we consider scheduling jobs with agreeable deadlines. We provide the first constant ratio competitive algorithm for the non-preemptive setting, which is of particular interest with regard to the known str…
We consider the online machine minimization problem in which jobs with hard deadlines arrive online over time at their release dates. The task is to determine a feasible schedule on a minimum number of machines. Our main result is a general O(m^2 log m)-competitive algorithm for the preemptive online problem, where m is the optimal number of machines used in an offline solution. This is the first improvement on an O(log (p_max/p_min))-competitive algorithm by Phillips et al. (STOC 1997), which was to date the best known algorithm even when m=2. Our algorithm is O(1)-competitive for any m that …
We propose a unifying framework based on configuration linear programs and randomized rounding, for different energy optimization problems in the dynamic speed-scaling setting. We apply our framework to various scheduling and routing problems in heterogeneous computing and networking environments. We first consider the energy minimization problem of scheduling a set of jobs on a set of parallel speed scalable processors in a fully heterogeneous setting. For both the preemptive-non-migratory and the preemptive-migratory variants, our approach allows us to obtain solutions of almost the same qua…
We investigate the problem of scheduling the maintenance of edges in a network, motivated by the goal of minimizing outages in transportation or telecommunication networks. We focus on maintaining connectivity between two nodes over time; for the special case of path networks, this is related to the problem of minimizing the busy time of machines. We show that the problem can be solved in polynomial time in arbitrary networks if preemption is allowed. If preemption is restricted to integral time points, the problem is NP-hard and in the non-preemptive case we give strong non-approximability re…
We consider online preemptive scheduling of jobs arriving one by one, to be assigned to two identical machines, with the goal of makespan minimization. We study the effect of selecting the best solution out of two independent solutions constructed in parallel in an online fashion. Two cases are analyzed, where one case is purely online, and in the other one jobs are presented sorted by non-increasing sizes. We show that using two solutions rather than one improves the performance significantly, but that an optimal solution cannot be obtained for any constant number of solutions constructed in …
Online scheduling has been a well studied and challenging research problem over the last five decades since the pioneering work of Graham with immense practical significance in various applications such as interactive parallel processing, routing in communication networks, distributed data management, client-server communications, traffic management in transportation, industrial manufacturing and production. In this problem, a sequence of jobs is received one by one in order by the scheduler for scheduling over a number of machines. On arrival of a job, the scheduler assigns the job irrevocabl…
The online Dial-a-Ride problem is a fundamental online problem in a metric space, where transportation requests appear over time and may be served in any order by a single server with unit speed. Restricted to the real line, online Dial-a-Ride captures natural problems like controlling a personal elevator. Tight results in terms of competitive ratios are known for the general setting and for online TSP on the line (where source and target of each request coincide). In contrast, online Dial-a-Ride on the line has resisted tight analysis so far, even though it is a very natural online problem. W…
This paper considers scheduling on identical machines. The scheduling objective considered in this paper generalizes most scheduling minimization problems. In the problem, there are $n$ jobs and each job $j$ is associated with a monotonically increasing function $g_j$. The goal is to design a schedule that minimizes $\sum_{j \in [n]} g_{j}(C_j)$ where $C_j$ is the completion time of job $j$ in the schedule. An $O(1)$-approximation is known for the single machine case. On multiple machines, this paper shows that if the scheduler is required to be either non-migratory or non-preemptive then any …
When designing a preemptive online algorithm for the maximum matching problem, we wish to maintain a valid matching M while edges of the underlying graph are presented one after the other. When presented with an edge e, the algorithm should decide whether to augment the matching M by adding e (in which case e may be removed later on) or to keep M in its current form without adding e (in which case e is lost for good). The objective is to eventually hold a matching M with maximum weight. The main contribution of this paper is to establish new lower and upper bounds on the competitive ratio achi…
We revisit a classical scheduling model to incorporate modern trends in data center networks and cloud services. Addressing some key challenges in the allocation of shared resources to user requests (jobs) in such settings, we consider the following variants of the classic {\em resource allocation problem} (\textsf{RAP}). The input to our problems is a set $J$ of jobs and a set $M$ of homogeneous hosts, each has an available amount of some resource. A job is associated with a release time, a due date, a weight, and a given length, as well as its resource requirement. A \emph{feasible} schedule…
The problem of scheduling with testing in the framework of explorable uncertainty models environments where some preliminary action can influence the duration of a task. In the model, each job has an unknown processing time that can be revealed by running a test. Alternatively, jobs may be run untested for the duration of a given upper limit. Recently, Dürr et al. [5] have studied the setting where all testing times are of unit size and have given lower and upper bounds for the objectives of minimizing the sum of completion times and the makespan on a single machine. In this paper, we extend t…
We investigate online scheduling with commitment for parallel identical machines. Our objective is to maximize the total processing time of accepted jobs. As soon as a job has been submitted, the commitment constraint forces us to decide immediately whether we accept or reject the job. Upon acceptance of a job, we must complete it before its deadline $d$ that satisfies $d \geq (1+ε)\cdot p + r$, with $p$ and $r$ being the processing time and the submission time of the job, respectively while $ε>0$ is the slack of the system. Since the hard case typically arises for near-tight deadlines, we con…
Scheduling with testing is a recent online problem within the framework of explorable uncertainty motivated by environments where some preliminary action can influence the duration of a task. Jobs have an unknown processing time that can be explored by running a test. Alternatively, jobs can be executed for the duration of a given upper limit. We consider this problem within the setting of multiple identical parallel machines and present competitive deterministic algorithms and lower bounds for the objective of minimizing the makespan of the schedule. In the non-preemptive setting, we present …
Shortest Remaining Processing Time (SRPT) is a well known preemptive scheduling algorithm for uniprocessor and multiprocessor systems. SRPT finds applications in the emerging areas such as scheduling of client's requests that are submitted to a web server for accessing static web pages, managing the access requests to files in multiuser database systems and routing of packets across several links as per bandwidth availability in data communications. SRPT has been proved to be optimal for the settings, where the objective is to minimize the mean response time of a list of jobs. According to our…
We consider the problem of scheduling $n$ jobs to minimize the makespan on $m$ unrelated machines, where job $j$ requires time $p_{ij}$ if processed on machine $i$. A classic algorithm of Lenstra et al. yields the best known approximation ratio of $2$ for the problem. Improving this bound has been a prominent open problem for over two decades. In this paper we obtain a tighter bound for a wide subclass of instances which can be identified efficiently. Specifically, we define the feasibility factor of a given instance as the minimum fraction of machines on which each job can be processed. We sh…
Structural properties of optimal preemptive schedules have been studied in a number of recent papers with a primary focus on two structural parameters: the minimum number of preemptions necessary, and a tight lower bound on `shifts', i.e., the sizes of intervals bounded by the times created by preemptions, job starts, or completions. So far only rough bounds for these parameters have been derived for specific problems. This paper sharpens the bounds on these structural parameters for a well-known open problem in the theory of preemptive scheduling: Instances consist of in-trees of $n$ unit-exe…
Makespan minimization on identical parallel machines is a classical scheduling problem. We consider the online scenario where a sequence of $n$ jobs has to be scheduled non-preemptively on $m$ machines so as to minimize the maximum completion time of any job. The best competitive ratio that can be achieved by deterministic online algorithms is in the range $[1.88,1.9201]$. Currently no randomized online algorithm with a smaller competitiveness is known, for general $m$. In this paper we explore the power of job migration, i.e.\ an online scheduler is allowed to perform a limited number of job …
We bound the performance guarantees that follow from Turán-like bounds for unweighted and weighted independent sets in bounded-degree graphs. In particular, a randomized approach of Boppana forms a simple 1-round distributed algorithm, as well as a streaming and preemptive online algorithm. We show it gives a tight $(Δ+1)/2$-approximation in unweighted graphs of maximum degree $Δ$, which is best possible for 1-round distributed algorithms. For weighted graphs, it gives only a $Δ$-approximation, but a simple modification results in an asymptotic expected $0.529 Δ$-approximation. This compares w…
We consider fundamental scheduling problems motivated by energy issues. In this framework, we are given a set of jobs, each with a release time, deadline and required processing length. The jobs need to be scheduled on a machine so that at most g jobs are active at any given time. The duration for which a machine is active (i.e., "on") is referred to as its active time. The goal is to find a feasible schedule for all jobs, minimizing the total active time. When preemption is allowed at integer time points, we show that a minimal feasible schedule already yields a 3-approximation (and this boun…
We consider the non-preemptive scheduling problem on identical machines where there is a parameter B and each machine in every unit length time interval can process up to B different jobs. The goal function we consider is the makespan minimization and we develop an EPTAS for this problem. Prior to our work a PTAS was known only for the case of one machine and constant values of B, and even the case of non-constant values of B on one machine was not known to admit a PTAS.
We prove that, for any arbitrary finite alphabet and for the uniform distribution over deterministic and accessible automata with n states, the average complexity of Moore's state minimization algorithm is in O(n log n). Moreover this bound is tight in the case of unary utomata.
We consider a variant of the NP-hard problem of assigning jobs to machines to minimize the completion time of the last job. Usually, precedence constraints are given by a partial order on the set of jobs, and each job requires all its predecessors to be completed before it can start. In his seminal paper, Graham (1966) presented a simple 2-approximation algorithm, and, more than 40 years later, Svensson (2010) proved that 2 is essentially the best approximation ratio one can hope for in general. In this paper, we consider a different type of precedence relation that has not been discussed as e…
The problem considered is the non-preemptive scheduling of independent jobs that consume a resource (which is non-renewable and replenished regularly) on parallel uniformly related machines. The input defines the speed of machines, size of jobs, the quantity of resource required by the jobs, the replenished quantities, and replenishment dates of the resource. Every job can start processing only after the required quantity of the resource is allocated to the job. The objective function is the minimization of the convex combination of the makespan and an objective that is equivalent to the $l_p$…
We investigate deterministic non-preemptive online scheduling with delayed commitment for total completion time minimization on parallel identical machines. In this problem, jobs arrive one-by-one and their processing times are revealed upon arrival. An online algorithm can assign a job to a machine at any time after its arrival. We neither allow preemption nor a restart of the job, that is, once started, the job occupies the assigned machine until its completion. Our objective is the minimization of the sum of the completion times of all jobs. In the more general weighted version of the probl…
We study discrepancy minimization for vectors in $\mathbb{R}^n$ under various settings. The main result is the analysis of a new simple random process in multiple dimensions through a comparison argument. As corollaries, we obtain bounds which are tight up to logarithmic factors for several problems in online vector balancing posed by Bansal, Jiang, Singla, and Sinha (STOC 2020), as well as linear time algorithms for logarithmic bounds for the Komlós conjecture.
We study policies aiming to minimize the weighted sum of completion times of jobs in the context of coordination mechanisms for selfish scheduling problems. Our goal is to design local policies that achieve a good price of anarchy in the resulting equilibria for unrelated machine scheduling. To obtain the approximation bounds, we introduce a new technique that while conceptually simple, seems to be quite powerful. With this method we are able to prove the following results. First, we consider Smith's Rule, which orders the jobs on a machine in ascending processing time to weight ratio, and sho…
We consider basic problems of non-preemptive scheduling on uniformly related machines. For a given schedule, defined by a partition of the jobs into m subsets corresponding to the m machines, C_i denotes the completion time of machine i. Our goal is to find a schedule which minimizes or maximizes \sum_{i=1}^m C_i^p for a fixed value of p such that 0<p<\infty. For p>1 the minimization problem is equivalent to the well-known problem of minimizing the \ell_p norm of the vector of the completion times of the machines, and for 0<p<1 the maximization problem is of interest. Our main result is an eff…
Google Research（谷歌研究院）；MIT（麻省理工学院）；UC Berkeley（加州大学伯克利分校）
We give a deterministic nearly-linear time algorithm for approximating any point inside a convex polytope with a sparse convex combination of the polytope's vertices. Our result provides a constructive proof for the Approximate Carathéodory Problem, which states that any point inside a polytope contained in the $\ell_p$ ball of radius $D$ can be approximated to within $ε$ in $\ell_p$ norm by a convex combination of only $O\left(D^2 p/ε^2\right)$ vertices of the polytope for $p \geq 2$. We also show that this bound is tight, using an argument based on anti-concentration for the binomial distrib…
In this article, we discuss an exact algorithm for solving mixed integer concave minimization problems. A piecewise inner-approximation of the concave function is achieved using an auxiliary linear program that leads to a bilevel program, which provides a lower bound to the original problem. The bilevel program is reduced to a single level formulation with the help of Karush-Kuhn-Tucker (KKT) conditions. Incorporating the KKT conditions lead to complementary slackness conditions that are linearized using BigM, for which we identify a tight value for general problems. Multiple bilevel programs,…
In this paper we study the limitations of parallelization in convex optimization. A convenient approach to study parallelization is through the prism of \emph{adaptivity} which is an information theoretic measure of the parallel runtime of an algorithm [BS18]. Informally, adaptivity is the number of sequential rounds an algorithm needs to make when it can execute polynomially-many queries in parallel at every round. For combinatorial optimization with black-box oracle access, the study of adaptivity has recently led to exponential accelerations in parallel runtime and the natural question is w…
The paper considers single-machine scheduling problems with a non-renewable resource. In this setting, we are given a set jobs, each of which is characterized by a processing time, a weight, and the job also has some resource requirement. At fixed points in time, a certain amount of the resource is made available to be consumed by the jobs. The goal is to assign the jobs non-preemptively to time slots on the machine, so that at any time their resource requirement does not exceed the available amounts of resources. The objective that we consider here is the minimization of the sum of weighted c…
We investigate the maximum-entropy model $\mathcal{B}_{n,m,p}$ for random $n$-vertex, $m$-edge multi-hypergraphs with expected edge size $pn$. We show that the expected size of the minimization of $\mathcal{B}_{n,m,p}$, i.e., the number of its inclusion-wise minimal edges, undergoes a phase transition with respect to $m$. If $m$ is at most $1/(1-p)^{(1-p)n}$, then the minimization is of size $Θ(m)$. Beyond that point, for $α$ such that $m = 1/(1-p)^{αn}$ and $\mathrm{H}$ being the entropy function, it is $Θ(1) \cdot \min\!\left(1, \, \frac{1}{(α\,{-}\,(1-p)) \sqrt{(1\,{-}\,α) n}}\right) \cdot …
For many computational problems involving randomness, intricate geometric features of the solution space have been used to rigorously rule out powerful classes of algorithms. This is often accomplished through the lens of the multi Overlap Gap Property ($m$-OGP), a rigorous barrier against algorithms exhibiting input stability. In this paper, we focus on the algorithmic tractability of two models: (i) discrepancy minimization, and (ii) the symmetric binary perceptron (\texttt{SBP}), a random constraint satisfaction problem as well as a toy model of a single-layer neural network. Our first focu…
