Online capacity maximization in wireless networks

The peak-demand charge motivates large-load customers to flatten their demand curves, while their self-owned renewable generations aggravate demand fluctuations. Thus, it is attractive to utilize energy storage for shaping real-time loads and reducing electricity bills. In this paper, we propose the first peak-aware competitive online algorithm for leveraging stored energy (e.g., in fuel cells) to minimize peak-demand charges. Our algorithm decides the discharging quantity slot by slot to maintain the optimal worst-case performance guarantee (namely, competitive ratio) among all deterministic online algorithms. Interestingly, we show that the best competitive ratio can be computed by solving a linear number of linear-fractional problems. We can also extend our competitive algorithm and analysis to improve the average-case performance and consider short-term prediction.

We develop a new approach for online network design and obtain improved competitive ratios for several problems. Our approach gives natural deterministic algorithms and simple analyses. At the heart of our work is a novel application of embeddings into hierarchically well-separated trees (HSTs) to the analysis of online network design algorithms --- we charge the cost of the algorithm to the cost of the optimal solution on any HST embedding of the terminals. This analysis technique is widely applicable to many problems and gives a unified framework for online network design. In a sense, our work brings together two of the main approaches to online network design. The first uses greedy-like algorithms and analyzes them using dual-fitting. The second uses tree embeddings and results in randomized $O(\log n)$-competitive algorithms, where $n$ is the total number of vertices in the graph. Our approach uses deterministic greedy-like algorithms but analyzes them via HST embeddings of the terminals. Our proofs are simpler as we do not need to carefully construct dual solutions and we get $O(\log k)$ competitive ratios, where $k$ is the number of terminals. In this paper, we apply our approach to obtain deterministic $O(\log k)$-competitive online algorithms for the following problems. - Steiner network with edge duplication. Previously, only a randomized $O(\log n)$-competitive algorithm was known. - Rent-or-buy. Previously, only deterministic $O(\log^2 k)$-competitive and randomized $O(\log k)$-competitive algorithms by Awerbuch, Azar and Bartal (2004) were known. - Connected facility location. Previously, only a randomized $O(\log^2 k)$-competitive algorithm by San Felice, Williamson and Lee (2014) was known. - Prize-collecting Steiner forest. We match the competitive ratio first achieved by Qian and Williamson (2011) and give a simpler analysis.

We study the k -server problem in the resource augmentation setting, i.e., when the performance of the online algorithm with k servers is compared to the offline optimal solution with h ≤ k servers. The problem is very poorly understood beyond uniform metrics. For this special case, the classic k -server algorithms are roughly (1+1/ϵ)-competitive when k =(1+ϵ) h , for any ϵ > 0. Surprisingly, however, no o ( h )-competitive algorithm is known even for HSTs of depth 2 and even when k / h is arbitrarily large. We obtain several new results for the problem. First, we show that the known k -server algorithms do not work even on very simple metrics. In particular, the Double Coverage algorithm has competitive ratio Ω ( h ) irrespective of the value of k , even for depth-2 HSTs. Similarly, the Work Function Algorithm, which is believed to be optimal for all metric spaces when k = h , has competitive ratio Ω ( h ) on depth-3 HSTs even if k =2 h . Our main result is a new algorithm that is O (1)-competitive for constant depth trees, whenever k =(1+ϵ) h for any ϵ > 0. Finally, we give a general lower bound that any deterministic online algorithm has competitive ratio at least 2.4 even for depth-2 HSTs and when k / h is arbitrarily large. This gives a surprising qualitative separation between uniform metrics and depth-2 HSTs for the ( h , k )-server problem.

We consider the following scheduling problem. The input is a set of jobs with equal processing times, where each job is specified by its release time and deadline. The goal is to determine a single-processor nonpreemptive schedule that maximizes the number of completed jobs. In the online version, each job arrives at its release time. We give two online algorithms with competitive ratios below $2$ and show several lower bounds on the competitive ratios. First, we give a barely random $5/3$-competitive algorithm that uses only one random bit. We also show a lower bound of $3/2$ on the competitive ratio of barely random algorithms that randomly choose one of two deterministic algorithms. If the two algorithms are selected with equal probability, we can further improve the bound to $8/5$. Second, we give a deterministic $3/2$-competitive algorithm in the model that allows restarts, and we show that in this model the ratio $3/2$ is optimal. For randomized algorithms with restarts we show a lower bound of $6/5$.

Online makespan minimization is a classical problem in which a sequence of jobs $$\sigma = J_1, \ldots , J_n$$ź=J1,ź,Jn has to be scheduled on m identical parallel machines so as to minimize the maximum completion time of any job. In this paper we investigate the problem with an essentially new model of resource augmentation. More specifically, an online algorithm is allowed to build several schedules in parallel while processing $$\sigma $$ź. At the end of the scheduling process the best schedule is selected. This model can be viewed as providing an online algorithm with extra space, which is invested to maintain multiple solutions. The setting is of particular interest in parallel processing environments where each processor can maintain a single or a small set of solutions. As a main result we develop a $$(4/3+\varepsilon )$$(4/3+ź)-competitive algorithm, for any $$0<\varepsilon \le 1$$0<ź≤1, that uses a constant number of schedules. The constant is equal to $$1/\varepsilon ^{O(\log (1/\varepsilon ))}$$1/źO(log(1/ź)). We also give a $$(1+\varepsilon )$$(1+ź)-competitive algorithm, for any $$0<\varepsilon \le 1$$0<ź≤1, that builds a polynomial number of $$(m/\varepsilon )^{O(\log (1/\varepsilon ) / \varepsilon )}$$(m/ź)O(log(1/ź)/ź) schedules. This value depends on m but is independent of the input $$\sigma $$ź. The performance guarantees are nearly best possible. We show that any algorithm that achieves a competitiveness smaller than 4 / 3 must construct $$\Omega (m)$$Ω(m) schedules. Our algorithms make use of novel guessing schemes that (1) predict the optimum makespan of a job sequence $$\sigma $$ź to within a factor of $$1+\varepsilon $$1+ź and (2) guess the job processing times and their frequencies in $$\sigma $$ź. In (2) we have to sparsify the universe of all guesses so as to reduce the number of schedules to a constant. We remark that the competitive ratios achieved using parallel schedules are considerably smaller than those in the standard problem without resource augmentation. Furthermore they are at least as good and in most cases better than the ratios obtained with other means of resource augmentation for makespan minimization.

Cloud providers such as Amazon EC2 and Google GCE typically price their idle compute resources at a significant discount in the form of transient servers. Unlike regular servers, cloud transient servers have no reliability guarantee and can be revoked anytime. In order to enjoy the price premium of transient servers while mitigating the impact of indefinite server revocations, cloud users continuously checkpoint current computational states to stable storage. To determine the optimal checkpointing interval, prior works have largely built upon predicting future revocations. However, we argue that existing prediction models may not work well if providers employ different server revocation algorithms in the future, and the required historical information can be unavailable in practice. In this work, we propose two online algorithms, one competitive and the other heuristic, that determine the checkpointing intervals without predicting future revocation based on history. Our competitive online algorithm achieves the best possible competitive ratio against the optimal, yet impractical, offline solution. Our heuristic algorithm adaptively reacts to server revocations, achieving better average performance than the competitive algorithm, especially for short-running tasks. Extensive trace-driven simulations confirm our analytical results and demonstrate the efficacy of the two online algorithms in the context of Amazon EC2 Spot Instances.

In this paper, we propose a general framework for online scheduling problems in which each machine has multiple states that lead to different processing times. For these problems, in addition to deciding how to assign jobs to machines, we also need to set the states of the machines each time they are assigned jobs. For a wide range of machine environments, job processing characteristics and constraints, and cost functions, we develop a ‐competitive deterministic online algorithm and a ‐competitive randomized online algorithm. The online weighted traveling repairman problem belongs to this general framework, and both our deterministic and randomized online algorithms lead to lower competitive ratios than the current existing ones in the literature. In addition, we include a complete proof that the online algorithm (reoptimizing the route of the repairman whenever a new request is released) is almost surely asymptotically optimal for a probabilistic version of this problem.

In this paper, we deal with competitive local on-line algorithms for non-preemptive channel allocation in mobile networks. The signal interferences in a network are modeled using an interference graph G. We prove that the greedy on-line algorithm is Δ-competitive, where Δ is the maximum degree of G. We employ the “classify and randomly select” paradigm ([5, 12]), and give a 5-competitive randomized algorithm for the case of planar interference graphs. We show that the problem of call control in mobile networks with multiple available frequencies reduces to the problem of call control in networks with a single frequency. Using this reduction, we present on-line algorithms for general networks with a single frequency. We give a local on-line algorithm which is \(\frac{{\alpha (\delta + 1 + \alpha )}}{{(1/2 + \alpha )^2 }}\)-competitive, where a is the independence number of G and δ is the average degree of G. The above results hold in the case that the duration of each request is infinite, and the benefit the algorithm gains by accepting each request is equal to 1. They are extended to handle requests of arbitrary durations, and arbitrary benefits. To our knowledge, this is the first time that competitive analysis is employed to study on-line algorithms for the channel allocation problem in mobile networks.

We study the admission control problem in general networks. Communication requests arrive over time, and the online algorithm accepts or rejects each request while maintaining the capacity limitations of the network. The admission control problem has been usually analyzed as a benefit problem, where the goal is to devise an online algorithm that accepts the maximum number of requests possible. The problem with this objective function is that even algorithms with optimal competitive ratios may reject almost all of the requests, when it would have been possible to reject only a few. This could be inappropriate for settings in which rejections are intended to be rare events.In this paper, we consider preemptive online algorithms whose goal is to minimize the number of rejected requests. Each request arrives together with the path it should be routed on. We show an O(log2(mc))-competitive randomized algorithm for the weighted case, where m is the number of edges in the graph and c is the maximum edge capacity. For the unweighted case, we give an O(log m log c)-competitive randomized algorithm. This settles an open question of Blum, Kalai and Kleinberg raised in [10]. We note that allowing preemption and handling requests with given paths are essential for avoiding trivial lower bounds.The admission control problem is a generalization of the online set cover with repetitions problem, whose input is a family of m subsets of a ground set of n elements. Elements of the ground set are given to the online algorithm one by one, possibly requesting each element a multiple number of times. (If each element arrives at most once, this corresponds to the online set cover problem.) The algorithm must cover each element by different subsets, according to the number of times it has been requested.We give an O(log m log n)-competitive randomized algorithm for the online set cover with repetitions problem. This matches a recent lower bound of Ω(log m log n) given by Feige and Korman for the competitive ratio of any randomized polynomial time algorithm, under the BPP ≠ NP assumption. Given any constant e > 0, an O(log m logn)-competitive deterministic bicriteria algorithm is shown that covers each element by at least (1-e)k sets, where k is the number of times the element is covered by the optimal solution.

We consider the classic online scheduling problem of minimizing the total weighted flow time on a single machine with preemptions. Here, each job j has an arbitrary arrival time rj, weight wj and size pj, and given a schedule its flow time is defined as the duration of time since its arrival until it completes its service requirement. The first non-trivial algorithms with poly-logarithmic competitive ratio for this problem were obtained relatively recently, and it was widely believed that the problem admits a constant factor competitive algorithm. In this paper, we show an ω(1) lower bound on the competitive ratio of any deterministic online algorithm. Our result is based on a gap amplification technique for online algorithms. Starting with a trivial lower bound of 1, we give a procedure to improve the lower bound sequentially, while ensuring at each step that the size of the instance increases relatively modestly.

Previous chapter Next chapter Full AccessProceedings Proceedings of the 2009 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)Weighted flow time does not admit O(1)-competitive algorithmsNikhil Bansal and Ho-Leung ChanNikhil BansalIBM T. J. Watson Research CenterMax-Planck-Institut für InformatikSearch for more papers by this author and Ho-Leung ChanIBM T. J. Watson Research CenterMax-Planck-Institut für InformatikSearch for more papers by this authorpp.1238 - 1244Chapter DOI:https://doi.org/10.1137/1.9781611973068.134PDFBibTexSections ToolsAdd to favoritesDownload CitationsTrack CitationsEmail SectionsAboutAbstract We consider the classic online scheduling problem of minimizing the total weighted flow time on a single machine with preemptions. Here, each job j has an arbitrary arrival time rj, weight wj and size pj, and given a schedule its flow time is defined as the duration of time since its arrival until it completes its service requirement. The first non-trivial algorithms with poly-logarithmic competitive ratio for this problem were obtained relatively recently, and it was widely believed that the problem admits a constant factor competitive algorithm. In this paper, we show an ω(1) lower bound on the competitive ratio of any deterministic online algorithm. Our result is based on a gap amplification technique for online algorithms. Starting with a trivial lower bound of 1, we give a procedure to improve the lower bound sequentially, while ensuring at each step that the size of the instance increases relatively modestly. Previous chapter Next chapter RelatedDetails Published:2009ISBN:978-0-89871-680-1eISBN:978-1-61197-306-8 https://doi.org/10.1137/1.9781611973068Book Series Name:ProceedingsBook Code:PR132Book Pages:xviii + 1288

We first show that for any bipartite graph H with at most five vertices there exists an on-line competitive algorithm for the class of H-free bipartite graphs. We then analyze the performance of an on-line algorithm for coloring bipartite graphs on various subfamilies. The algorithm yields new upper bounds for the on-line chromatic number of bipartite graphs. We prove that the algorithm is on-line competitive for $P_7$-free bipartite graphs, i.e., that do not contain an induced path on seven vertices. The number of colors used by the on-line algorithm for $P_6$-free and $P_7$-free bipartite graphs is, respectively, bounded by roughly twice and roughly eight times the on-line chromatic number. In contrast, it is known that there exists no competitive on-line algorithm to color $P_6$-free (or $P_7$-free) bipartite graphs, i.e., for which the number of colors is bounded by any function depending only on the chromatic number.

We study the advice complexity and the random bit complexity of the online knapsack problem: Given a knapsack of unit capacity, and n items that arrive in successive time steps, an online algorithm has to decide for every item whether it gets packed into the knapsack or not. The goal is to maximize the value of the items in the knapsack without exceeding its capacity. In the model of advice complexity of online problems, one asks how many bits of advice about the unknown parts of the input are both necessary and sufficient to achieve a specific competitive ratio. It is well-known that even the unweighted online knapsack problem does not admit any competitive deterministic online algorithm. We show that a single bit of advice helps a deterministic algorithm to become 2-competitive, but that Ω(log n) advice bits are necessary to further improve the deterministic competitive ratio. This is the first time that such a phase transition for the number of advice bits has been observed for any problem. We also show that, surprisingly, instead of an advice bit, a single random bit allows for a competitive ratio of 2, and any further amount of randomness does not improve this. Moreover, we prove that, in a resource augmentation model, i.e., when allowing a little overpacking of the knapsack, a constant number of advice bits suffices to achieve a near-optimal competitive ratio. We also study the weighted version of the problem proving that, with O(log n) bits of advice, we can get arbitrarily close to an optimal solution and, using asymptotically fewer bits, we are not competitive.

This paper addresses the classic online Steiner tree problem on edge-weighted graphs. It is known that a greedy (nearest neighbor) online algorithm has a tight competitive ratio for wide classes of graphs, such as trees, rings, any class including series-parallel graphs, and unweighted graphs with bounded diameter. However, we do not know any greedy or non-greedy tight deterministic algorithm for other classes of graphs. In this paper, we observe that a greedy algorithm is $$\Omega (\log n)$$ -competitive on outerplanar graphs, where n is the number of vertices, and propose a 5.828-competitive deterministic algorithm on outerplanar graphs. Our algorithm connects a requested vertex and the tree constructed thus far using a path that is constant times longer than the distance between them. We also present a lower bound of 4 for arbitrary deterministic online Steiner tree algorithms on outerplanar graphs.

We consider the online version of the piercing set problem, where geometric objects arrive one by one, and the online algorithm must maintain a valid piercing set for the already arrived objects by making irrevocable decisions. It is easy to observe that any deterministic algorithm solving this problem for intervals in R has a competitive ratio of at least Ω(n). This paper considers the piercing set problem for similarly sized objects. We propose a deterministic online algorithm for similarly sized fat objects in R d . For homothetic hypercubes in R d with side length in the range [1, k], we propose a deterministic algorithm having a competitive ratio of at most 3 d ⌈log 2 k⌉ + 2 d . In the end, we show deterministic lower bounds of the competitive ratio for similarly sized α-fat objects in R 2 and homothetic hypercubes in R d . Note that piercing translated copies of a convex object is equivalent to the unit covering problem, which is well-studied in the online setup. Surprisingly, no upper bound of the competitive ratio was known for the unit covering problem when the corresponding object is anything other than a ball or a hypercube. Our result yields an upper bound of the competitive ratio for the unit covering problem when the corresponding object is any convex object in R d .