Mobile edge computing is a new cloud computing paradigm, which makes use of small-sized edge clouds to provide real-time services to users. These mobile edge-clouds (MECs) are located in close proximity to users, thus enabling users to seamlessly access applications running on MECs. Due to the co-existence of the core (centralized) cloud, users, and one or multiple layers of MECs, an important problem is to decide where (on which computational entity) to place different components of an application. This problem, known as the application or workload placement problem, is notoriously hard, and therefore, heuristic algorithms without performance guarantees are generally employed in common practice, which may unknowingly suffer from poor performance as compared with the optimal solution. In this paper, we address the application placement problem and focus on developing algorithms with provable performance bounds. We model the user application as an application graph and the physical computing system as a physical graph, with resource demands availabilities annotated on these graphs. We first consider the placement of a linear application graph and propose an algorithm for finding its optimal solution. Using this result, we then generalize the formulation and obtain online approximation algorithms with polynomial-logarithmic (poly-log) competitive ratio for tree application graph placement. We jointly consider node and link assignment, and incorporate multiple types of computational resources at nodes.
One important aspect to note is that most existing work, including @cite_23 @cite_12 @cite_13 @cite_35 @cite_17 @cite_15 , do not specifically consider the online operation of the algorithms. Although some of them implicitly claim that one can apply the algorithm repeatedly for each newly arrived application, the competitive ratio of such procedure is unclear. To the best of our knowledge, @cite_34 is the only work that studied the competitive ratio of the online application placement problem that considers .
@cite_35 @cite_23 @cite_15 @cite_34 @cite_13 @cite_12 @cite_17
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In this paper, we study online algorithms that schedule malleable jobs, i.e., jobs that can be parallelized on any subset of the available @math m identical machines. We study a model that accounts for the tradeoff between multiprocessor speedup and overhead time, namely, if job @math j has processing requirement @math pj and is assigned to run on @math kj machines, then @math j's execution time becomes @math pj kj+(kj-1)c, where @math c is a constant parameter to the problem. For @math m=2, we present an online algorithm OCS that has a strong competitive ratio of 3 2, matching a previously established lower bound. We also present an online algorithm ASYM2 that is asymptotically @math ((4-∈) (3-∈))-competitive when @math m=2, where @math 0<∈≤2 is a parameter to the algorithm, improving upon an existing asymptotically (3 2)-competitive algorithm. Finally, we present an online algorithm OTO that is strongly @math 2-competitive when @math m=3, improving upon the previous best upper bound of @math 9 4. The problem considered here is the same as the one discussed in [G. Galambos and G. J. Woeginger, eds., SIAM J. Comput., 22 (1993), pp. 349--355]. It is an m-machine online scheduling problem in which we wish to minimize the competitive ratio for the makespan objective. In this paper, we show that @math is a lower bound on this competitive ratio for m=4. In particular, we show how to force a lower bound of @math for any positive @math . This reduces the gap between the performance of known algorithms [S. Albers, in Proceedings of the 29th Annual ACM Symposium on Theory of Computing, ACM, New York, 1997, pp. 130--139] and the lower bound. The method used introduces an approach to building the task master's strategy. In this paper we study the classic online matching problem, introduced in the seminal work of Karp, Vazirani and Vazirani (STOC 1990), in regular graphs. For such graphs, an optimal deterministic algorithm as well as efficient algorithms under stochastic input assumptions were known. In this work, we present a novel randomized algorithm with competitive ratio tending to one on this family of graphs, under adversarial arrival order. Our main contribution is a novel algorithm which achieves competitive ratio [EQUATION] in expectation on d-regular graphs. In contrast, we show that all previously-studied online algorithms have competitive ratio strictly bounded away from one. Moreover, we show the convergence rate of our algorithm's competitive ratio to one is nearly tight, as no algorithm achieves competitive ratio better than [EQUATION]. Finally, we show that our algorithm yields a similar competitive ratio with high probability, as well as guaranteeing each vertex a probability of being matched tending to one. We consider the problem of scheduling jobs that are given as groups of non-intersecting segments on the real line. Each job J j is associated with an interval, I j , which consists of up to t segments, for some t ≥ 1, a positive weight, w j , and two jobs are in conflict if any of their segments intersect. Such jobs show up in a wide range of applications, including the transmission of continuous-media data, allocation of linear resources (e.g. bandwidth in linear processor arrays), and in computational biology geometry. The objective is to schedule a subset of non-conflicting jobs of maximum total weight.In a single machine environment, our problem can be formulated as the problem of finding a maximum weight independent set in a t-interval graph (the special case of t = 1 is an ordinary interval graph). We show that, for t ≥ 2, this problem is APX-hard, even for highly restricted instances. Our main result is a 2t-approximation algorithm for general instances, based on a novel fractional version of the Local Ratio technique. Previously, the problem was considered only for proper union graphs, a restricted subclass of t-interval graphs, and the approximation factor achieved was (2t - 1 + 1 2t). A bi-criteria polynomial time approximation scheme (PTAS) is developed for the subclass of t-union graphs.In the online case, we consider uniform weight jobs that consist of at most two segments. We show that when the resulting 2-interval graph is proper, a simple greedy algorithm is 3-competitive, while any randomized algorithm has competitive ratio at least 2.5. For general instances, we give a randomized O(log2 R)-competitive (or O((log R)2+e)-competitive) algorithm, where R is the known (unknown) ratio between the longest and the shortest segment in the input sequence.