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.
Some other related work focuses on graph partitioning, such as @cite_13 and @cite_35 , where the physical graph is defined as a complete graph with edge costs associated with the distance or latency between physical servers. Such an abstraction combines multiple network links into one (abstract) physical edge, which may hide the actual status of individual links along a path.
@cite_35 @cite_13
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We consider the problem of partitioning the nodes of a graph with costs on its edges into subsets of given sizes so as to minimize the sum of the costs on all edges cut. This problem arises in several physical situations — for example, in assigning the components of electronic circuits to circuit boards to minimize the number of connections between boards. This paper presents a heuristic method for partitioning arbitrary graphs which is both effective in finding optimal partitions, and fast enough to be practical in solving large problems. Vertex-centric graph processing systems such as Pregel, PowerGraph, or GraphX recently gained popularity due to their superior performance of data analytics on graph-structured data. These systems exploit the graph structure to improve data access locality during computation, making use of specialized graph partitioning algorithms. Recent partitioning techniques assume a uniform and constant amount of data exchanged between graph vertices (i.e., uniform vertex traffic) and homogeneous underlying network costs. However, in real-world scenarios vertex traffic and network costs are heterogeneous. This leads to suboptimal partitioning decisions and inefficient graph processing. To this end, we designed GrapH, the first graph processing system using vertex-cut graph partitioning that considers both, diverse vertex traffic and heterogeneous network, to minimize overall communication costs. The main idea is to avoid frequent communication over expensive network links using an adaptive edge migration strategy. Our evaluations show an improvement of 60 in communication costs compared to state-of-the-art partitioning approaches. In this paper we study the use of spectral techniques for graph partitioning. Let G = (V, E) be a graph whose vertex set has a ‘latent’ partition V1,. . ., Vk. Moreover, consider a ‘density matrix’ Ɛ = (Ɛvw)v, sw∈V such that, for v ∈ Vi and w ∈ Vj, the entry Ɛvw is the fraction of all possible Vi−Vj-edges that are actually present in G. We show that on input (G, k) the partition V1,. . ., Vk can (very nearly) be recovered in polynomial time via spectral methods, provided that the following holds: Ɛ approximates the adjacency matrix of G in the operator norm, for vertices v ∈ Vi, w ∈ Vj ≠ Vi the corresponding column vectors Ɛv, Ɛw are separated, and G is sufficiently ‘regular’ with respect to the matrix Ɛ. This result in particular applies to sparse graphs with bounded average degree as n = #V → ∞, and it has various consequences on partitioning random graphs. Graph partitioning is one of the most studied NP-complete problems. Given a graph G=(V, E) , the task is to partition the vertex set V into k disjoint subsets of about the same size, such that the number of edges with endpoints in different subsets is minimized. In this paper, we present a highly effective multilevel memetic algorithm, which integrates a new multiparent crossover operator and a powerful perturbation-based tabu search algorithm. The proposed crossover operator tends to preserve the backbone with respect to a certain number of parent individuals, i.e., the grouping of vertices which is common to all parent individuals. Extensive experimental studies on numerous benchmark instances from the graph partitioning archive show that the proposed approach, within a time limit ranging from several minutes to several hours, performs far better than any of the existing graph partitioning algorithms in terms of solution quality.