The cloud computing industry has grown rapidly over the last decade, and with this growth there is a significant increase in demand for compute resources. Demand is manifested in the form of Virtual Machine (VM) requests, which need to be assigned to physical machines in a way that minimizes resource fragmentation and efficiently utilizes the available machines. This problem can be modeled as a dynamic version of the bin packing problem with the objective of minimizing the total usage time of the bins (physical machines). Earlier works on dynamic bin packing assumed that no knowledge is available to the scheduler and later works studied models in which lifetime/duration of each "item" (VM in our context) is available to the scheduler. This extra information was shown to improve exponentially the achievable competitive ratio.
Motivated by advances in Machine Learning that provide good estimates of workload characteristics, this paper studies the effect of having extra information regarding future (total) demand. In the cloud context, since demand is an aggregate over many VM requests, it can be predicted with high accuracy (e.g., using historical data). We show that the competitive factor can be dramatically improved by using this additional information; in some cases, we achieve constant competitiveness, or even a competitive factor that approaches 1. Along the way, we design new offline algorithms with improved approximation ratios for the dynamic bin-packing problem.
Inspired by online ad allocation, we study online stochastic packing linear programs from theoretical and practical standpoints. We first present a near-optimal online algorithm for a general class of packing linear programs which model various online resource allocation problems including online variants of routing, ad allocations, generalized assignment, and combinatorial auctions. As our main theoretical result, we prove that a simple primal-dual training-based algorithm achieves a (1 - o(1))-approximation guarantee in the random order stochastic model. This is a significant improvement ove…
We present a unified framework for designing and analyzing algorithms for online budgeted allocation problems (including online matching) and their generalization, the Online Generalized Assignment Problem (OnGAP). These problems have been intensively studied as models of how to allocate impressions for online advertising. In contrast to previous analyses of online budgeted allocation algorithms (the so-called "balance" or "water-filling" family of algorithms) our analysis is based on the method of randomized dual fitting, analogous to the recent analysis of the RANKING algorithm for online ma…
We consider the problem of online allocation (matching and assortments) of reusable resources where customers arrive sequentially in an adversarial fashion and allocated resources are used or rented for a stochastic duration that is drawn independently from known distributions. Focusing on the case of large inventory, we give an algorithm that is $(1-1/e)$ competitive for general usage distributions. At the heart of our result is the notion of a relaxed online algorithm that is only subjected to fluid approximations of the stochastic elements in the problem. The output of this algorithm serves…
For online resource allocation problems, we propose a new demand arrival model where the sequence of arrivals contains both an adversarial component and a stochastic one. Our model requires no demand forecasting; however, due to the presence of the stochastic component, we can partially predict future demand as the sequence of arrivals unfolds. Under the proposed model, we study the problem of the online allocation of a single resource to two types of customers, and design online algorithms that outperform existing ones. Our algorithms are adjustable to the relative size of the stochastic comp…
We address the question of whether the primal-dual approach for the design and analysis of online algorithms can be applied to nonmonotone problems. We provide a positive answer by presenting a primal-dual analysis to the online algorithm of Awerbuch et al.[AAPW01] for routing virtual circuits with unknown durations.
We study the following vertex-weighted online bipartite matching problem: $G(U, V, E)$ is a bipartite graph. The vertices in $U$ have weights and are known ahead of time, while the vertices in $V$ arrive online in an arbitrary order and have to be matched upon arrival. The goal is to maximize the sum of weights of the matched vertices in $U$. When all the weights are equal, this reduces to the classic \emph{online bipartite matching} problem for which Karp, Vazirani and Vazirani gave an optimal $\left(1-\frac{1}{e}\right)$-competitive algorithm in their seminal work~\cite{KVV90}. Our main resu…
We consider an online allocation problem that involves a set $P$ of $n$ players and a set $E$ of $m$ indivisible entities over discrete time steps $1,2,\ldots,τ$. At each time step $t \in [1,τ]$, for every entity $e \in E$, there is a restriction list $L_t(e)$ that prescribes the subset of players to whom $e$ can be assigned and a non-negative value $v_t(e,p)$ of $e$ to every player $p \in P$. The sets $P$ and $E$ are fixed beforehand. The sets $L_t(\cdot)$ and values $v_t(\cdot,\cdot)$ are given in an online fashion. An allocation is a distribution of $E$ among $P$, and we are interested in t…
In this paper, we study a general online linear programming problem whose formulation encompasses many practical dynamic resource allocation problems, including internet advertising display applications, revenue management, various routing, packing, and auction problems. We propose a model, which under mild assumptions, allows us to design near-optimal learning-based online algorithms that do not require the a priori knowledge about the total number of online requests to come, a first of its kind. We then consider two variants of the problem that relax the initial assumptions imposed on the pr…
We study the following problem, introduced by Chung et al. in 2006. We are given, online or offline, a set of coloured items of different sizes, and wish to pack them into bins of equal size so that we use few bins in total (at most $α$ times optimal), and that the items of each colour span few bins (at most $β$ times optimal). We call such allocations $(α, β)$-approximate. As usual in bin packing problems, we allow additive constants and consider $(α,β)$ as the asymptotic performance ratios. We prove that for $\eps>0$, if we desire small $α$, no scheme can beat $(1+\eps, Ω(1/\eps))$-approxima…
Southeast University（东南大学）；University of Victoria（维多利亚大学）；New Jersey Institute of Technology（新泽西理工学院）
With the popularity of the Internet, traditional offline resource allocation has evolved into a new form, called online resource allocation. It features the online arrivals of agents in the system and the real-time decision-making requirement upon the arrival of each online agent. Both offline and online resource allocation have wide applications in various real-world matching markets ranging from ridesharing to crowdsourcing. There are some emerging applications such as rebalancing in bike sharing and trip-vehicle dispatching in ridesharing, which involve a two-stage resource allocation proce…
We study the performance of a best reply algorithm for online resource allocation problems with a diseconomy of scale. In an online resource allocation problem, we are given a set of resources and a set of requests that arrive in an online manner. Each request consists of a set of feasible allocations and an allocation is a set of resources. The total cost of an allocation vector is given by the sum of the resources' costs, where each resource's cost depends on the total load on the resource under the allocation vector. We analyze the natural online procedure where each request is allocated gr…
We show that the popular water-filling algorithm for maximizing the mutual information in parallel Gaussian channels is sub-modular. The sub-modularity of water-filling algorithm is then used to derive online basestation allocation algorithms, where mobile users are assigned to one of many possible basestations immediately and irrevocably upon arrival without knowing the future user information. The goal of the allocation is to maximize the sum-rate of the system under power allocation at each basestation. We present online algorithms with competitive ratio of at most 2 when compared to offlin…
We consider the following online allocation problem: Given a unit square S, and a sequence of numbers n_i between 0 and 1, with partial sum bounded by 1; at each step i, select a region C_i of previously unassigned area n_i in S. The objective is to make these regions compact in a distance-aware sense: minimize the maximum (normalized) average Manhattan distance between points from the same set C_i. Related location problems have received a considerable amount of attention; in particular, the problem of determining the "optimal shape of a city", i.e., allocating a single n_i has been studied. …
We study an online version of the max-min fair allocation problem for indivisible items. In this problem, items arrive one by one, and each item must be allocated irrevocably on arrival to one of $n$ agents, who have additive valuations for the items. Our goal is to make the least happy agent as happy as possible. In research on the topic of online allocation, this is a fundamental and natural problem. Our main result is to reveal the asymptotic competitive ratios of the problem for both the adversarial and i.i.d. input models. We design a polynomial-time deterministic algorithm that is asympt…
Modern Primal-Dual Frameworks for Prior-Free Online Resource Allocation
University of Chicago Booth School of Business（芝加哥大学布斯商学院）；Department of Industrial Engineering & Operations Research, University of California, Berkeley（加州大学伯克利分校工业工程与运筹学系）
Motivated by Internet advertising applications, online allocation problems have been studied extensively in various adversarial and stochastic models. While the adversarial arrival models are too pessimistic, many of the stochastic (such as i.i.d or random-order) arrival models do not realistically capture uncertainty in predictions. A significant cause for such uncertainty is the presence of unpredictable traffic spikes, often due to breaking news or similar events. To address this issue, a simultaneous approximation framework has been proposed to develop algorithms that work well both in the…
Design of {\it online algorithms} for assigning mobile users to basestations is considered with the objective of maximizing the sum-rate, when all users associated to any one basestation equally share each basestation's resources. Each user on its arrival reveals the rates it can obtain if connected to each of the basestations, and the problem is to assign each user to any one basestation irrevocably so that the sum-rate is maximized at the end of all user arrivals, without knowing the future user arrival or rate information or its statistics at each user arrival. Online algorithms with consta…
We provide a reduction from revenue maximization to welfare maximization in multi-dimensional Bayesian auctions with arbitrary (possibly combinatorial) feasibility constraints and independent bidders with arbitrary (possibly combinatorial) demand constraints, appropriately extending Myerson's result to this setting. We also show that every feasible Bayesian auction can be implemented as a distribution over virtual VCG allocation rules. A virtual VCG allocation rule has the following simple form: Every bidder's type t_i is transformed into a virtual type f_i(t_i), via a bidder-specific function…
The paper investigates a version of the resource allocation problem arising in the wireless networking, namely in the OVSF code reallocation process. In this setting a complete binary tree of a given height $n$ is considered, together with a sequence of requests which have to be served in an online manner. The requests are of two types: an insertion request requires to allocate a complete subtree of a given height, and a deletion request frees a given allocated subtree. In order to serve an insertion request it might be necessary to move some already allocated subtrees to other locations in or…
We consider a new and general online resource allocation problem, where the goal is to maximize a function of a positive semidefinite (PSD) matrix with a scalar budget constraint. The problem data arrives online, and the algorithm needs to make an irrevocable decision at each step. Of particular interest are classic experiment design problems in the online setting, with the algorithm deciding whether to allocate budget to each experiment as new experiments become available sequentially. We analyze two greedy primal-dual algorithms and provide bounds on their competitive ratios. Our analysis re…
School of Operations Research and Information Engineering, Cornell University（康奈尔大学运营研究与信息工程学院）
We develop a framework for designing simple and efficient policies for a family of online allocation and pricing problems, that includes online packing, budget-constrained probing, dynamic pricing, and online contextual bandits with knapsacks. In each case, we evaluate the performance of our policies in terms of their regret (i.e., additive gap) relative to an offline controller that is endowed with more information than the online controller. Our framework is based on Bellman Inequalities, which decompose the loss of an algorithm into two distinct sources of error: (1) arising from computatio…
Cainiao Network（菜鸟网络）；City University of Hong Kong（香港城市大学）；University of Science and Technology of China（中国科学技术大学）
In this paper, we investigate the online allocation problem of maximizing the overall revenue subject to both lower and upper bound constraints. Compared to the extensively studied online problems with only resource upper bounds, the two-sided constraints affect the prospects of resource consumption more severely. As a result, only limited violations of constraints or pessimistic competitive bounds could be guaranteed. To tackle the challenge, we define a measure of feasibility $ξ^*$ to evaluate the hardness of this problem, and estimate this measurement by an optimization routine with theoret…
Columbia University（哥伦比亚大学）；University of Chicago（芝加哥大学）；Google Research（谷歌研究院）
Online allocation problems with resource constraints are central problems in revenue management and online advertising. In these problems, requests arrive sequentially during a finite horizon and, for each request, a decision maker needs to choose an action that consumes a certain amount of resources and generates reward. The objective is to maximize cumulative rewards subject to a constraint on the total consumption of resources. In this paper, we consider a data-driven setting in which the reward and resource consumption of each request are generated using an input model that is unknown to t…
Online Resource Allocation with Convex-set Machine-Learned Advice
Sloan School of Management, Massachusetts Institute of Technology（麻省理工学院斯隆管理学院）；Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology（麻省理工学院电气工程与计算机科学系）；Department of Industrial Engineering and Decision Analytics, Hong Kong University of Science and Technology（香港科学与技术大学工业工程与决策分析系）
Can one choose a good Huffman code on the fly, without knowing the underlying distribution? Online Slot Allocation (OSA) models this and similar problems: There are n slots, each with a known cost. There are n items. Requests for items are drawn i.i.d. from a fixed but hidden probability distribution p. After each request, if the item, i, was not previously requested, then the algorithm (knowing the slot costs and the requests so far, but not p) must place the item in some vacant slot j(i). The goal is to minimize the sum, over the items, of the probability of the item times the cost of its as…
Learning-Augmented Competitive Algorithms for Spatiotemporal Online Allocation with Deadline Constraints
University of Massachusetts Amherst（马萨诸塞大学阿默斯特分校）；California Institute of Technology（加州理工学院）；University of Waterloo（滑铁卢大学）；Massachusetts Institute of Technology（麻省理工学院）
We improve the best known competitive ratio (from 1/4 to 1/2), for the online multi-unit allocation problem, where the objective is to maximize the single-price revenue. Moreover, the competitive ratio of our algorithm tends to 1, as the bid-profile tends to ``smoothen''. This algorithm is used as a subroutine in designing truthful auctions for the same setting: the allocation has to be done online, while the payments can be decided at the end of the day. Earlier, a reduction from the auction design problem to the allocation problem was known only for the unit-demand case. We give a reduction …
