Online Stochastic Matching: Beating 1-1/e

We perform an experimental study of algorithms for online bipartite matching under the known i.i.d input model with integral types. In the last decade, there has been substantial effort in designing complex algorithms to improve worst-case approximation ratios. Our goal is to determine how these algorithms perform on more practical instances rather than worst-case instances. In particular, we are interested in whether the ranking of the algorithms by their worst-case performance is consistent with the ranking of the algorithms by their average-case/practical performance. We are also interested in whether preprocessing times and implementation difficulties that are introduced by these algorithms are justified in practice. To that end, we evaluate these algorithms on different random inputs as well as real-life instances obtained from publicly available repositories. We compare these algorithms against several simple greedy-style algorithms. Most of the complex algorithms in the literature are presented as being non-greedy (i.e., an algorithm can intentionally skip matching a node that has available neighbors) to simplify the analysis. Every such algorithm can be turned into a greedy one without hurting its worst-case performance. On our benchmarks, non-greedy versions of these algorithms perform much worse than their greedy versions. Greedy versions perform about as well as the simplest greedy algorithm by itself. This, together with our other findings, suggests that simplest greedy algorithms are competitive with the state-of-the-art worst-case algorithms for online bipartite matching on many average-case and practical input families. Greediness is by far the most important property of online algorithms for bipartite matching.

We consider variants of the online stochastic bipartite matching problem motivated by Internet advertising display applications, as introduced in Feldman et al. [Feldman J, Mehta A, Mirrokni VS, Muthukrishnan S (2009) Online stochastic matching: Beating 1 − 1/e. FOCS '09: Proc. 50th Annual IEEE Sympos. Foundations Comput. Sci. (IEEE, Washington, DC), 117–126]. In this setting, advertisers express specific interests into requests for impressions of different types. Advertisers are fixed and known in advance, whereas requests for impressions come online. The task is to assign each request to an interested advertiser (or to discard it) immediately upon its arrival. In the adversarial online model, the ranking algorithm of Karp et al. [Karp RM, Vazirani UV, Varirani VV (1990) An optimal algorithm for online bipartite matching. STOC '90: Proc. 22nd Annual ACM Sympos. Theory Comput. (ACM, New York), 352–358] provides a best possible randomized algorithm with competitive ratio 1 − 1/e ≈ 0.632. In the stochastic i.i.d. model, when requests are drawn repeatedly and independently from a known probability distribution over the different impression types, Feldman et al. [Feldman J, Mehta A, Mirrokni VS, Muthukrishnan S (2009) Online stochastic matching: Beating 1 − 1/e. FOCS '09: Proc. 50th Annual IEEE Sympos. Foundations Comput. Sci. (IEEE, Washington, DC), 117–126] prove that one can do better than 1 − 1/e. Under the restriction that the expected number of request of each impression type is an integer, they provide a 0.670-competitive algorithm, later improved by Bahmani and Kapralov [Bahmani B, Kapralov M (2010) Improved bounds for online stochastic matching. ESA '10: Proc. 22nd Annual Eur. Sympos. Algorithms (Springer-Verlag, Berlin, Heidelberg), 170–181] to 0.699 and by Manshadi et al. [Manshadi V, Gharan SO, Saberi A (2012) Online stochastic matching: Online actions based on offline statistics. Math. Oper. Res. 37(4):559–573] to 0.705. Without this integrality restriction, Manshadi et al. are able to provide a 0.702-competitive algorithm. In this paper we consider a general class of online algorithms for the i.i.d. model that improve on all these bounds and that use computationally efficient offline procedures (based on the solution of simple linear programs of maximum flow types). Under the integrality restriction on the expected number of impression types, we get a 1 − 2e−2(≈0.729)-competitive algorithm. Without this restriction, we get a 0.706-competitive algorithm. Our techniques can also be applied to other related problems such as the online stochastic vertex-weighted bipartite matching problem as defined in Aggarwal et al. [Aggarwal G, Goel G, Karande C, Mehta A (2011) Online vertex-weighted bipartite matching and single-bid budgeted allocations. SODA '11: Proc. 22nd Annual ACM-SIAM Sympos. Discrete Algorithms (SIAM, Philadelphia), 1253–1264]. For this problem, we obtain a 0.725-competitive algorithm under the stochastic i.i.d. model with integral arrival rate. Finally, we show the validity of all our results under a Poisson arrival model, removing the need to assume that the total number of arrivals is fixed and known in advance, as is required for the analysis of the stochastic i.i.d. models described above.

We propose a general method for converting online algorithms to local computation algorithms, by selecting a random permutation of the input, and simulating running the online algorithm. We bound the number of steps of the algorithm using a query tree, which models the dependencies between queries. We improve previous analyses of query trees on graphs of bounded degree, and extend this improved analysis to the cases where the degrees are distributed binomially, and to a special case of bipartite graphs. Using this method, we give a local computation algorithm for maximal matching in graphs of bounded degree, which runs in time and space O(log3 n). We also show how to convert a large family of load balancing algorithms (related to balls and bins problems) to local computation algorithms. This gives several local load balancing algorithms which achieve the same approximation ratios as the online algorithms, but run in O(logn) time and space. Finally, we modify existing local computation algorithms for hypergraph 2-coloring and k-CNF and use our improved analysis to obtain better time and space bounds, of O(log4 n), removing the dependency on the maximal degree of the graph from the exponent.

Energy efficient algorithms are becoming critically important, as huge data centers and server farms have increasing impact on monetary and environmental costs. Motivated by such issues, we study online load balancing from an energy perspective. Our framework extends recent work by Khuller, Li, and Saha (SODA 2010) to the online model. We are given m machines, each with some energy activation cost c i and d dimensions (i.e., components). There are n jobs which arrive online and must be assigned to machines. Each job induces a load on its assigned machine along each dimension. We must select machines to activate so that the total activation cost of the machines falls within a budget B and the largest load over all machines and dimensions (i.e., the makespan) by assigning jobs to active machines is at most Λ.We first study the model in which machines are unrelated and can have arbitrary activation cost. In this problem, which we call Machine Activation, we extend previous work to handle jobs which arrive online. We consider a variant where the target makespan Λ and budget B are given. The first main result is an online algorithm which is O(log(md) log(nm))-competitive on the load Λ and O(dlog2 (nm))-competitive on the energy budget B. We also address cases where one parameter is given and we are asked to minimize the other, or where we want to minimize a convex combination of the two. Running our previous algorithm in phases gives results for these variants. We prove lower bounds indicating that the effect on the competitive ratio due to multiple phases is necessary.Our second main result is in the same setting except all machines are identical and have no activation cost. We call this problem Vector Load Balancing, our objective is to minimize the largest load induced over all machines and dimensions (makespan) and the sum of the largest induced load on each machine (energy). We give an online algorithm that is O(logd)-competitive on makespan, which improves even on the best prior offline result, and O(logd)-competitive on the sum of induced loads if the target makespan is given; without this knowledge we show that it is impossible to get a competitive ratio independent of m.KeywordsVirtual MachineActivation CostCompetitive RatioOnline AlgorithmGeneralize Assignment ProblemThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We provide simple but surprisingly useful direct product theorems for proving lower bounds on online algorithms with a limited amount of advice about the future. Intuitively, our direct product theorems say that if b bits of advice are needed to ensure a cost of at most t for some problem, then r*b bits of advice are needed to ensure a total cost of at most r*t when solving r independent instances of the problem. Using our direct product theorems, we are able to translate decades of research on randomized online algorithms to the advice complexity model. Doing so improves significantly on the previous best advice complexity lower bounds for many online problems, or provides the first known lower bounds. For example, we show that - A paging algorithm needs Omega(n) bits of advice to achieve a competitive ratio better than H_k = Omega(log k), where k is the cache size. Previously, it was only known that Omega(n) bits of advice were necessary to achieve a constant competitive ratio smaller than 5/4. - Every O(n^{1-epsilon})-competitive vertex coloring algorithm must use Omega(n log n) bits of advice. Previously, it was only known that Omega(n log n) bits of advice were necessary to be optimal. For certain online problems, including the MTS, k-server, metric matching, paging, list update, and dynamic binary search tree problem, we prove that randomization and sublinear advice are equally powerful (if the underlying metric space or node set is finite). This means that several long-standing open questions regarding randomized online algorithms can be equivalently stated as questions regarding online algorithms with sublinear advice. For example, we show that there exists a deterministic O(log k)-competitive k-server algorithm with sublinear advice if and only if there exists a randomized O(log k)-competitive k-server algorithm without advice. Technically, our main direct product theorem is obtained by extending an information theoretical lower bound technique due to Emek, Fraigniaud, Korman, and Rosen [ICALP'09].

We study a variant of the offline Dial-a-Ride problem, where each request has a source and destination and the goal is to maximize the number of requests served within a specified time limit. We investigate this problem for the uniform metric space and show that the problem is NP-hard. We then present a 2/3 approximation algorithm called TWOCHAIN, which simply looks for pairs of requests that are “chained” together and serves those before serving requests that are not connected to any others. We also show that a natural generalization of this algorithm, k-chain, has an approximation ratio at most 7/9. We also analyze the longest-chain-first algorithm for the problem, characterizing graphs on which it is optimal, and showing that it has an approximation ratio no better than 5/6. Our experiments on all of these algorithms show that TWOCHAIN is a promising algorithm, performing nearly as well as more computationally intensive variants. We dedicate this article to the memory of Gerhard Woeginger, whose life and work greatly influenced our professional lives, as expanded upon in the Acknowledgments. Woeginger’s prolific research in scheduling, matching, bin-packing, TSP, and online algorithms in general, all served as important parts of the foundation on which our own scholarly pursuits were shaped and formed over the years. Woeginger also studied Dial-a-Ride (DARP) Problems, as DARP is a generalization both of scheduling problems and of TSP, which were two of his most active areas of research.

In this paper, the authors consider an on-line scheduling problem of m (m ≥ 3) identical machines with common maintenance time interval and nonresumable availability. For the case that the length of maintenance time interval is larger than the largest processing time of jobs, the authors prove that any on-line algorithm has not a constant competitive ratio. For the case that the length of maintenance time interval is less than or equal to the largest processing time of jobs, the authors prove a lower bound of 3 on the competitive ratio. The authors give an on-line algorithm with competitive ratio \(4 - \tfrac{1} {m} \). In particular, for the case of m = 3, the authors prove the competitive ratio of the on-line algorithm is \(\tfrac{{10}} {3} \).

The demand for cloud computing has increased manifold in the recent past. More specifically, on-demand computing has seen a rapid rise as organizations rely mostly on cloud service providers for their day-to-day computing needs. The cloud service provider fulfills different user requirements using virtualization - where a single physical machine can host multiple Virtual Machines. Each virtual machine potentially represents a different user environment such as operating system, programming environment, and applications. However, these cloud services use a large amount of electrical energy and produce greenhouse gases. To reduce the electricity cost and greenhouse gases, energy efficient algorithms must be designed. One specific area where energy efficient algorithms are required is virtual machine consolidation. With virtual machine consolidation, the objective is to utilize the minimum possible number of hosts to accommodate the required virtual machines, keeping in mind the service level agreement requirements. This research work formulates the virtual machine migration as an online problem and develops optimal offline and online algorithms for the single host virtual machine migration problem under a service level agreement constraint for an over-utilized host. The online algorithm is analyzed using a competitive analysis approach. In addition, an experimental analysis of the proposed algorithm on real-world data is conducted to showcase the improved performance of the proposed algorithm against the benchmark algorithms. Our proposed online algorithm consumed 25% less energy and performed 43% fewer migrations than the benchmark algorithms.

We consider a natural online optimization problem set on the real line. The state of the online algorithm at each integer time is a location on the real line. At each integer time, a convex function arrives online. In response, the online algorithm picks a new location. The cost paid by the online algorithm for this response is the distance moved plus the value of the function at the final destination. The objective is then to minimize the aggregate cost over all time. The motivating application is rightsizing power-proportional data centers. We give a 2-competitive algorithm for this problem. We also give a 3-competitive memoryless algorithm, and show that this is the best competitive ratio achievable by a deterministic memoryless algorithm. Finally we show that this online problem is strictly harder than the standard ski rental problem.

We survey how the advice complexity of online algorithms can be used to obtain lower bounds on the performance of randomized online algorithms. Online algorithms with advice may query an oracle that knows the whole input from the start to solve some instance of an online problem. This is done by reading a finite prefix of some infinite binary advice tape, which is created by the oracle before the first piece of input is processed. Similarly, a randomized online algorithm may use a binary tape where every bit is chosen uniformly at random.In this survey, we review a technique, similar to Yao’s principle, which allows statements on the advice complexity of some given online problem to translate to results on the power of randomization for this problem in terms of lower bounds. We give some examples where this technique works and how it is applied, and show its limitations and that it is tight in a very general sense.KeywordsCompetitive RatioOnline AlgorithmOnline ComputationInput LengthDeterministic StrategyThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this paper, we study a stochastic variant of the celebrated k-server problem. In the k-server problem, we are required to minimize the total movement of k servers that are serving an online sequence of t requests in a metric. In the stochastic setting we are given t independent distributions <P_1, P_2,..., P_t> in advance, and at every time step i a request is drawn from Pi. Designing the optimal online algorithm in such setting is NP-hard, therefore the emphasis of our work is on designing an approximately optimal online algorithm. We first show a structural characterization for a certain class of non-adaptive online algorithms. We prove that in general metrics, the best of such algorithms has a cost of no worse than three times that of the optimal online algorithm. Next, we present an integer program that finds the optimal algorithm of this class for any arbitrary metric. Finally, by rounding the solution of the linear relaxation of this program, we present an online algorithm for the stochastic k-server problem with the approximation factor of 3 in the line and circle metrics and O(log n) in a general metric of size n. Moreover, we define the Uber problem, in which each demand consists of two endpoints, a source and a destination. We show that given an a-approximation algorithm for the k-server problem, we can obtain an (a+2)-approximation algorithm for the Uber problem. Motivated by the fact that demands are usually highly correlated with the time we study the stochastic Uber problem. Furthermore, we extend our results to the correlated setting where the probability of a request arriving at a certain point depends not only on the time step but also on the previously arrived requests.

In this thesis, we study the theoretical approach on energy-efficient scheduling problems arising in demand response management in the modern electrical smart grid. Consumers send in power requests with flexible feasible timeslots 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. We study the smart grid scheduling problem in different models. For the offline model, we prove the problem is NP-hard for the general case. We propose a polynomial time algorithm for special input where jobs have unit power request and unit time duration. By adapting the polynomial time algorithm for unit-size jobs, we propose an approximation algorithm for more general input. On the other hand, we also present an exact algorithm to find the optimal schedule for the problem with general input. For the online model, we propose an online algorithm for jobs with jobs with arbitrary power request, arbitrary time duration, and arbitrary contiguous feasible intervals. We also show a lower bound of the competitive ratio for the smart grid scheduling problem with unit height and arbitrary width. For special cases, we design different online algorithms with better competitive ratios. Finally, we look at other optimization problems and show how to solve them by adapting our techniques. We prove that our online algorithm can solve the machine minimization problem with an asymptotically optimal competitive ratio. We also show that our exact algorithm can be adapted to solve other demand response management problems.

Delayed Information and Action in On-Line Algorithms

Objectives: Scheduling is an optimization problem in computer science and operation research in which ideal jobs are assigned to particular times. Nowadays, the problem is presented as an online problem, that is, each job is presented and the online algorithm needs to make a decision about the job before the next job is presented. Methods and Statstical Analysis: The proposed approach is used to solve the open job scheduling problem in which any job can be connected with the available machine. That was implemented in matlab to available best results with hybrid evolutionary algorithm. Findings: In this paper, a hybrid algorithm based on the particle swarm optimization is proposed, for flexible, open shop scheduling problem, to minimize the make-span. First an effective new approach using two decisions based on parallel priories dispatching rules is applied. Next we develop a hybridizing HPSO, that presents new components for updating velocity and position using evolutionary operators, with an adaptive neighbourhood procedure based on the insert-interchange fitness function, selection, mutation, crossover. Application/Improvements: The performance of the proposed a new hybrid algorithm is compared to other benchmark problems. Keywords: Dispatching Rules, Evolutionary Operators, Flexible Open Shop Problem, Local Search, Particle Swarm Optimization

In classical \acsmpc it is assumed that a fixed and a priori known set of parties wants to securely evaluate a function of their private inputs. This assumption implies that online problems, in which the set of parties that arrive and leave over time are not a priori known, are not covered by the classical setting. Therefore, the notion of online \acsmpc has been introduced, and a general feasibility result has been proven that shows that any online algorithm can be implemented as a distributed protocol that is secure in this setting~\citeklinger_towards_2021,klinger_corrected_towards_2022. However, so far, no online \acsmpc protocol that implements a concrete online algorithm has been proposed and evaluated such that the practicality of the constructive proof is an open question. We close this gap and propose the first privacy-preserving online \acsmpc protocol for the prominent problem of fully online matching with deadlines. In this problem an (a priori unknown) set of parties with their inputs arrive over time and can then be matched with other parties until they leave when their individual deadline is reached. We prove that our protocol is statistically secure in the presence of a semi-honest adversary that controls strictly less than half of the parties present at each point in time. We extensively evaluate the performance of our protocol in three different network settings, various input sizes and different matching conditions, as well as various numbers of parties.