The Virtual Network Embedding Problem (VNEP) captures the essence of many resource allocation problems of today’s infrastructure providers, which offer their physical computation and networking resources to customers. Customers request resources in the form of Virtual Networks, i.e. as a directed graph which specifies computational requirements at the nodes and communication requirements on the edges. An embedding of a Virtual Network on the shared physical infrastructure is the joint mapping of (virtual) nodes to physical servers together with the mapping of (virtual) edges onto paths in the physical network connecting the respective servers.This work initiates the study of approximation algorithms for the VNEP. Concretely, we study the offline setting with admission control: given multiple request graphs the task is to embed the most profitable subset while not exceeding resource capacities. Our approximation is based on the randomized rounding of Linear Programming (LP) solutions. Interestingly, we uncover that the standard LP formulation exhibits an inherent structural deficit when considering general virtual networks: its solutions cannot be decomposed into valid embeddings. In turn, focusing on the class of cactus request graphs, we devise a novel LP formulation, whose solutions can be decomposed into convex combinations of valid embedding. Proving performance guarantees of our rounding scheme, we obtain the first approximation algorithm for the VNEP in the resource augmentation model.We propose two rounding heuristics and evaluate their performance in an extensive computational study, showing that these consistently yield good solutions (even without augmentations).
Considering approximation algorithms, employed randomized rounding in @cite_7 to obtain a constant approximation for embedding line requests on arbitrary substrate graphs under strong assumptions on the benefits and the capacities. In their interesting work, @cite_11 give an @math time @math -approximation algorithm for minimizing the load of embedding @math -depth trees based on a @math -node substrate. Their result is based on a strong LP relaxation inspired by the Sherali-Adams hierarchy.
@cite_7 @cite_11
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We study a basic resource allocation problem that arises in cloud computing environments. The physical network of the cloud is represented as a graph with vertices denoting servers and edges corresponding to communication links. A workload is a set of processes with processing requirements and mutual communication requirements. The workloads arrive and depart over time, and the resource allocator must map each workload upon arrival to the physical network. We consider the objective of minimizing the congestion. We show that solving a subproblem about mapping a single workload to the physical graph essentially suffices to solve the general problem. In particular, an α-approximation for this single mapping problem gives an O(α log nD)-competitive algorithm for the general problem, where n is the number of nodes in the physical network and D is the maximum to minimum workload duration ratio. We also show how to solve the single mapping problem for two natural class of workloads, namely depth-d-trees and complete-graph workloads. For depth-d tree, we give an nO(d) time O(d2 log (nd))-approximation based on a strong LP relaxation inspired by the Sherali-Adams hierarchy. We present a simple distributed @math -approximation algorithm for maximum weight independent set (MaxIS) in the @math model which completes in @math rounds, where @math is the maximum degree, @math is the number of rounds needed to compute a maximal independent set (MIS) on @math , and @math is the maximum weight of a node. Whether our algorithm is randomized or deterministic depends on the MIS algorithm used as a black-box. Plugging in the best known algorithm for MIS gives a randomized solution in @math rounds, where @math is the number of nodes. We also present a deterministic @math -round algorithm based on coloring. We then show how to use our MaxIS approximation algorithms to compute a @math -approximation for maximum weight matching without incurring any additional round penalty in the @math model. We use a known reduction for simulating algorithms on the line graph while incurring congestion, but we show our algorithm is part of a broad family of for which we describe a mechanism that allows the simulation to run in the @math model without an additional overhead. Next, we show that for maximum weight matching, relaxing the approximation factor to ( @math ) allows us to devise a distributed algorithm requiring @math rounds for any constant @math . For the unweighted case, we can even obtain a @math -approximation in this number of rounds. These algorithms are the first to achieve the provably optimal round complexity with respect to dependency on @math . In this paper the shortest path and the minimum spanning tree problems in a graph with @math nodes and @math cost scenarios (objectives) are discussed. In order to choose a solution the min-max criterion is applied. The minmax versions of both problems are hard to approximate within @math for any @math . The best approximation algorithm for the min-max shortest path problem, known to date, has approximation ratio of @math . On the other hand, for the min-max spanning tree, there is a randomized algorithm with approximation ratio of @math . In this paper a deterministic @math -approximation algorithm for min-max shortest path is constructed. For min-max spanning tree a deterministic @math -approximation algorithm is proposed, which works for a large class of graphs and a randomized @math -approximation algorithm, which can be applied to all graphs, is constructed. It is also shown that the approximation ratios obtained are close to the integrality gaps of the corresponding LP relaxations. The subspace approximation problem Subspace( @math , @math ) asks for a @math -dimensional linear subspace that fits a given set of points optimally, where the error for fitting is a generalization of the least squares fit and uses the @math norm instead. Most of the previous work on subspace approximation has focused on small or constant @math and @math , using coresets and sampling techniques from computational geometry. In this paper, extending another line of work based on convex relaxation and rounding, we give a polynomial time algorithm, , with the approximation guarantee roughly @math , where @math is the @math -th moment of a standard normal random variable N(0,1). We show that the convex relaxation we use has an integrality gap (or "rank gap") of @math , for any constant @math . Finally, we show that assuming the Unique Games Conjecture, the subspace approximation problem is hard to approximate within a factor better than @math , for any constant @math . Many resource allocation problems in the cloud can be described as a basic Virtual Network Embedding Problem (VNEP): the problem of finding a mapping of a request graph (describing a workload) onto a substrate graph (describing the physical infrastructure). Applications range from mapping testbeds (from where the problem originated), over the embedding of batch-processing workloads (virtual clusters) to the embedding of service function chains. The different applications come with their own specific requirements and constraints, including node mapping constraints, routing policies, and latency constraints. While the VNEP has been studied intensively over the last years, complexity results are only known for specific models and we lack a comprehensive understanding of its hardness.This paper charts the complexity landscape of the VNEP by providing a systematic analysis of the hardness of a wide range of VNEP variants, using a unifying and rigorous proof framework. In particular, we show that the problem of finding a feasible embedding is @math -complete in general, and, hence, the VNEP cannot be approximated under any objective, unless @math holds. Importantly, we derive @math -completeness results also for finding approximate embeddings, which may violate, e.g., capacity constraints by certain factors. Lastly, we prove that our results still pertain when restricting the request graphs to planar or degree-bounded graphs. We develop approximation algorithms for the problem of placing replicated data in arbitrary networks, where the nodes may both issue requests for data objects and have capacity for storing data objects so as to minimize the average data-access cost. We introduce the data placement problem to model this problem. We have a set of caches @math , a set of clients @math , and a set of data objects @math . Each cache @math can store at most @math data objects. Each client @math has demand @math for a specific data object @math and has to be assigned to a cache that stores that object. Storing an object @math in cache @math incurs a storage cost of @math , and assigning client @math to cache @math incurs an access cost of @math . The goal is to find a placement of the data objects to caches respecting the capacity constraints, and an assignment of clients to caches so as to minimize the total storage and client access costs. We present a 10-approximation algorithm for this problem. Our algorithm is based on rounding an optimal solution to a natural linear-programming relaxation of the problem. One of the main technical challenges encountered during rounding is to preserve the cache capacities while incurring only a constant-factor increase in the solution cost. We also introduce the connected data placement problem to capture settings where write-requests are also issued for data objects, so that one requires a mechanism to maintain consistency of data. We model this by requiring that all caches containing a given object be connected by a Steiner tree to a root for that object, which issues a multicast message upon a write to (any copy of) that object. The total cost now includes the cost of these Steiner trees. We devise a 14-approximation algorithm for this problem. We show that our algorithms can be adapted to handle two variants of the problem: (a) a @math -median variant, where there is a specified bound on the number of caches that may contain a given object, and (b) a generalization where objects have lengths and the total length of the objects stored in any cache must not exceed its capacity. We present two new approximation algorithms for the problem of finding a k-node connected spanning subgraph (directed or undirected) of minimum cost. The best known approximation guarantees for this problem were @math for both directed and undirected graphs, and @math for undirected graphs with @math , where @math is the number of nodes in the input graph. Our first algorithm has approximation ratio @math , which is @math except for very large values of @math , namely, @math . This algorithm is based on a new result on @math -connected @math -critical graphs, which is of independent interest in the context of graph theory. Our second algorithm uses the primal-dual method and has approximation ratio @math for all values of @math . Combining these two gives an algorithm with approximation ratio @math , which asymptotically improves the best known approximation guarantee for directed graphs for all values of @math , and for undirected graphs for @math . Moreover, this is the first algorithm that has an approximation guarantee better than @math for all values of @math . Our approximation ratio also provides an upper bound on the integrality gap of the standard LP-relaxation. Several algorithms with an approximation guarantee of @math are known for the Set Cover problem, where @math is the number of elements. We study a generalization of the Set Cover problem, called the Partition Set Cover problem. Here, the elements are partitioned into @math , and we are required to cover at least @math elements from each color class @math , using the minimum number of sets. We give a randomized LP-rounding algorithm that is an @math approximation for the Partition Set Cover problem. Here @math denotes the approximation guarantee for a related Set Cover instance obtained by rounding the standard LP. As a corollary, we obtain improved approximation guarantees for various set systems for which @math is known to be sublogarithmic in @math . We also extend the LP rounding algorithm to obtain @math approximations for similar generalizations of the Facility Location type problems. Finally, we show that many of these results are essentially tight, by showing that it is NP-hard to obtain an @math -approximation for any of these problems. This paper considers stochastic optimization problems whose objective functions involve powers of random variables. For example, consider the classic Stochastic lp Load Balancing Problem (SLBp): There are @math machines and @math jobs, and known independent random variables @math decribe the load incurred on machine @math if we assign job @math to it. The goal is to assign each jobs to machines in order to minimize the expected @math -norm of the total load on the machines. While convex relaxations represent one of the most powerful algorithmic tools, in problems such as SLBp the main difficulty is to capture the objective function in a way that only depends on each random variable separately. We show how to capture @math -power-type objectives in such separable way by using the @math -function method, introduced by Lata a to relate the moment of sums of random variables to the individual marginals. We show how this quickly leads to a constant-factor approximation for very general subset selection problem with @math -moment objective. Moreover, we give a constant-factor approximation for SLBp, improving on the recent @math -approximation of [, SODA 18]. Here the application of the method is much more involved. In particular, we need to sharply connect the expected @math -norm of a random vector with the @math -moments of its marginals (machine loads), taking into account simultaneously the different scales of the loads that are incurred by an unknown assignment. We consider distributed algorithms for approximate maximum matching on general graphs. Our main result is a randomized @math -approximation distributed algorithm for maximum weighted matching, whose running time is @math for any constant @math , where @math is the number of nodes in the graph. This is, to the best of our knowledge, the first log-time distributed algorithm that achieves constant approximation for maximum weighted matching on general graphs. In addition, we consider the dynamic case, where nodes are inserted and deleted one at a time. For unweighted dynamic graphs, we give a distributed algorithm that maintains a @math -approximation in @math time for each node insertion or deletion for any constant @math . For weighted dynamic graphs we give a constant-factor approximation distributed algorithm that runs in constant time for each insertion or deletion. In the capacitated facility location problem with hard capacities, we are given a set of facilities, @math , and a set of clients @math in a common metric space. Each facility i has a facility opening cost f i and capacity u i that specifies the maximum number of clients that may be assigned to this facility. We want to open some facilities from the set @math and assign each client to an open facility so that at most u i clients are assigned to any open facility i. The cost of assigning client j to facility i is given by the distance c ij , and our goal is to minimize the sum of the facility opening costs and the client assignment costs. The only known approximation algorithms that deliver solutions within a constant factor of optimal for this NP-hard problem are based on local search techniques. It is an open problem to devise an approximation algorithm for this problem based on a linear programming lower bound (or indeed, to prove a constant integrality gap for any LP relaxation). We make progress on this question by giving a 5-approximation algorithm for the special case in which all of the facility costs are equal, by rounding the optimal solution to the standard LP relaxation. One notable aspect of our algorithm is that it relies on partitioning the input into a collection of single-demand capacitated facility location problems, approximately solving them, and then combining these solutions in a natural way. Many resource allocation problems in the cloud can be described as a basic Virtual Network Embedding Problem (VNEP): finding mappings of request graphs (describing the workloads) onto a substrate graph (describing the physical infrastructure). In the offline setting, the two natural objectives are profit maximization, i.e., embedding a maximal number of request graphs subject to the resource constraints, and cost minimization, i.e., embedding all requests at minimal overall cost. The VNEP can be seen as a generalization of classic routing and call admission problems, in which requests are arbitrary graphs whose communication endpoints are not fixed. Due to its applications, the problem has been studied intensively in the networking community. However, the underlying algorithmic problem is hardly understood. This paper presents the first fixed-parameter tractable approximation algorithms for the VNEP. Our algorithms are based on randomized rounding. Due to the flexible mapping options and the arbitrary request graph topologies, we show that a novel linear program formulation is required. Only using this novel formulation the computation of convex combinations of valid mappings is enabled, as the formulation needs to account for the structure of the request graphs. Accordingly, to capture the structure of request graphs, we introduce the graph-theoretic notion of extraction orders and extraction width and show that our algorithms have exponential runtime in the request graphs' maximal width. Hence, for request graphs of fixed extraction width, we obtain the first polynomial-time approximations. Studying the new notion of extraction orders we show that (i) computing extraction orders of minimal width is NP-hard and (ii) that computing decomposable LP solutions is in general NP-hard, even when restricting request graphs to planar ones. This paper addresses the basic question of how well a tree can approximate distances of a metric space or a graph. Given a graph, the problem of constructing a spanning tree in a graph which strongly preserves distances in the graph is a fundamental problem in network design. We present scaling distortion embeddings where the distortion scales as a function of @math , with the guarantee that for each @math simultaneously, the distortion of a fraction @math of all pairs is bounded accordingly. Quantitatively, we prove that any finite metric space embeds into an ultrametric with scaling distortion @math . For the graph setting, we prove that any weighted graph contains a spanning tree with scaling distortion @math . These bounds are tight even for embedding into arbitrary trees. These results imply that the average distortion of the embedding is constant and that the @math distortion is @math . For probabilistic embedding into spanning trees we prove a scaling distortion of @math , which implies constant @math -distortion for every fixed @math . Modern cloud infrastructure providers allow customers to rent computing capability in the form of a network of virtual machines (VMs) with bandwidth guarantees between pairs of VMs. Typical requests are in the form of a chain of VMs with an uplink bandwidth to the gateway node of the network (rooted path requests), and most data center architectures route network packets along a spanning tree of the physical network. VMs are instantiated inside servers which reside at the leaves of this network, leading to the following optimization problem: given a rooted tree network T and a set of rooted path requests, find an embedding of the requests that minimizes link congestion. Our main result is an algorithm that, given a rooted tree network T with n leaves and set of weighted rooted path requests, embeds a 1−e fraction of the requests with congestion at most poly(logn, logθ,e−1)·OPT (approximation is necessary since the problem is NP-hard). Here OPT is the congestion of the optimal embedding and θ is the ratio of the maximum to minimum weights of the path requests. We also obtain an O(Hlogn e2) approximation if node capacities can be augmented by a (1+e) factor (here H is the height of the tree). Our algorithm applies a randomized rounding scheme based on Group Steiner Tree rounding to a novel LP relaxation of the set of subtrees of T with a given number of leaves that may be of independent interest. One of the classic results in scheduling theory is the @math -approximation algorithm by Lenstra, Shmoys, and Tardos for the problem of scheduling jobs to minimize makespan on unrelated machines; i.e., job @math requires time @math if processed on machine @math . More than two decades after its introduction it is still the algorithm of choice even in the restricted model where processing times are of the form @math . This problem, also known as the restricted assignment problem, is NP-hard to approximate within a factor less than @math , which is also the best known lower bound for the general version. Our main result is a polynomial time algorithm that estimates the optimal makespan of the restricted assignment problem within a factor @math , where @math is an arbitrarily small constant. The result is obtained by upper bounding the integrality gap of a certain strong linear program, known as the configuration LP, that was previously successfu... We present a near-optimal distributed algorithm for @math -approximation of single-commodity maximum flow in undirected weighted networks that runs in @math communication rounds in the model. Here, @math and @math denote the number of nodes and the network diameter, respectively. This is the first improvement over the trivial bound of @math , and it nearly matches the @math round complexity lower bound. The development of the algorithm contains two results of independent interest: (i) A @math -round distributed construction of a spanning tree of average stretch @math . (ii) A @math -round distributed construction of an @math -congestion approximator consisting of the cuts induced by @math virtual trees. The distributed representation of the cut approximator allows for evaluation in @math rounds. All our algorithms make use of randomization and succeed with high probability. The min-sum k-clustering problem in a metric space is to find a partition of the space into k clusters as to minimize the total sum of distances between pairs of points assigned to the same cluster. We give the first polynomial time non-trivial approximation algorithm for this problem. The algorithm provides an @math approximation to the min-sum k -clustering problem in general metric spaces, with running time @math . The result is based on embedding of metric spaces into hierarchically separated trees. We also provide a bicriteria approximation result that provides a constant approximation factor solution with only a constant factor increase in the number of clusters. This result is obtained by modifying and drawing ideas from recently developed primal dual approximation algorithms for facility location. We give an approximation algorithm for non-uniform sparsest cut with the following guarantee: For any @math , given cost and demand graphs with edge weights @math respectively, we can find a set @math with @math at most @math times the optimal non-uniform sparsest cut value, in time @math provided @math . Here @math is the @math 'th smallest generalized eigenvalue of the Laplacian matrices of cost and demand graphs; @math (resp. @math ) is the weight of edges crossing the @math cut in cost (resp. demand) graph and @math is the sparsity of the optimal cut. In words, we show that the non-uniform sparsest cut problem is easy when the generalized spectrum grows moderately fast. To the best of our knowledge, there were no results based on higher order spectra for non-uniform sparsest cut prior to this work. Even for uniform sparsest cut, the quantitative aspects of our result are somewhat stronger than previous methods. Similar results hold for other expansion measures like edge expansion, normalized cut, and conductance, with the @math 'th smallest eigenvalue of the normalized Laplacian playing the role of @math in the latter two cases. Our proof is based on an l1-embedding of vectors from a semi-definite program from the Lasserre hierarchy. The embedded vectors are then rounded to a cut using standard threshold rounding. We hope that the ideas connecting @math -embeddings to Lasserre SDPs will find other applications. Another aspect of the analysis is the adaptation of the column selection paradigm from our earlier work on rounding Lasserre SDPs [GS11] to pick a set of edges rather than vertices. This feature is important in order to extend the algorithms to non-uniform sparsest cut. Clustering is a fundamental tool in data mining. It partitions points into groups (clusters) and may be used to make decisions for each point based on its group. However, this process may harm protected (minority) classes if the clustering algorithm does not adequately represent them in desirable clusters -- especially if the data is already biased. At NIPS 2017, proposed a model for fair clustering requiring the representation in each cluster to (approximately) preserve the global fraction of each protected class. Restricting to two protected classes, they developed both a 4-approximation for the fair @math -center problem and a @math -approximation for the fair @math -median problem, where @math is a parameter for the fairness model. For multiple protected classes, the best known result is a 14-approximation for fair @math -center. We extend and improve the known results. Firstly, we give a 5-approximation for the fair @math -center problem with multiple protected classes. Secondly, we propose a relaxed fairness notion under which we can give bicriteria constant-factor approximations for all of the classical clustering objectives @math -center, @math -supplier, @math -median, @math -means and facility location. The latter approximations are achieved by a framework that takes an arbitrary existing unfair (integral) solution and a fair (fractional) LP solution and combines them into an essentially fair clustering with a weakly supervised rounding scheme. In this way, a fair clustering can be established belatedly, in a situation where the centers are already fixed. We consider @math -median clustering in finite metric spaces and @math -means clustering in Euclidean spaces, in the setting where @math is part of the input (not a constant). For the @math -means problem, show that if the optimal @math -means clustering of the input is more expensive than the optimal @math -means clustering by a factor of @math , then one can achieve a @math -approximation to the @math -means optimal in time polynomial in @math and @math by using a variant of Lloyd's algorithm. In this work we substantially improve this approximation guarantee. We show that given only the condition that the @math -means optimal is more expensive than the @math -means optimal by a factor @math for some constant @math , we can obtain a PTAS. In particular, under this assumption, for any @math we achieve a @math -approximation to the @math -means optimal in time polynomial in @math and @math , and exponential in @math and @math . We thus decouple the strength of the assumption from the quality of the approximation ratio. We also give a PTAS for the @math -median problem in finite metrics under the analogous assumption as well. For @math -means, we in addition give a randomized algorithm with improved running time of @math . Our technique also obtains a PTAS under the assumption of that all @math approximations are @math -close to a desired target clustering, in the case that all target clusters have size greater than @math and @math is constant. Note that the motivation of is that for many clustering problems, the objective function is only a proxy for the true goal of getting close to the target. From this perspective, our improvement is that for @math -means in Euclidean spaces we reduce the distance of the clustering found to the target from @math to @math when all target clusters are large, and for @math -median we improve the largeness'' condition needed in the work of to get exactly @math -close from @math to @math . Our results are based on a new notion of clustering stability.