We study the online facility location problem with uniform facility costs in the random-order model. Meyerson's algorithm [FOCS'01] is arguably the most natural and simple online algorithm for the problem with several advantages and appealing properties. Its analysis in the random-order model is one of the cornerstones of random-order analysis beyond the secretary problem. Meyerson's algorithm was shown to be (asymptotically) optimal in the standard worst-case adversarial-order model and 8-competitive in the random order model. While this bound in the random-order model is the long-standing state-of-the-art, it is not known to be tight, and the true competitive-ratio of Meyerson's algorithm remained an open question for more than two decades. We resolve this question and prove tight bounds on the competitive-ratio of Meyerson's algorithm in the random-order model, showing that it is exactly 4-competitive. Following our tight analysis, we introduce a generic parameterized version of Meyerson's algorithm that retains all the advantages of the original version. We show that the best algorithm in this family is exactly 3-competitive. On the other hand, we show that no online algorithm for this problem can achieve a competitive-ratio better than 2. Finally, we prove that the algorithms in this family are robust to partial adversarial arrival orders.

We study the problem of maximizing a monotone submodular function subject to a Multiple Knapsack constraint. The input is a set I of items, each has a non-negative weight, and a set of bins of arbitrary capacities. Also, we are given a submodular, monotone and non-negative function f over subsets of the items. The objective is to find a packing of a subset of items A⊆I in the bins such that f(A) is maximized. Our main result is an almost optimal polynomial time (1−e−1−ε)-approximation algorithm for the problem, for any ε>0. The algorithm relies on a structuring technique which converts a general multiple knapsack constraint to a constraint in which the bins are partitioned into groups of exponentially increasing cardinalities, each consisting of bins of uniform capacity. We derive the result by combining structuring with a refined analysis of techniques for submodular optimization subject to knapsack constraints.

Cloud computing is transforming networking landscape over the last few years. The first order of business for major cloud providers today is to attract as many organizations as possible to their own clouds. To that end cloud providers offer a new generation of managed network solutions to connect the premises of the enterprises to their clouds. To serve their customers better and to innovate fast, major cloud providers are currently on the route to building their own "private Internets", which are idiosyncratic. On the other hand, customers that do not want to stay locked by vendors and who want flexibility in using best-for-the-task services spanning multiple clouds and, possibly, their own premises, seek for solutions that will provide smart overlay connectivity across clouds. The result of these developments is a multiplication of closed idiosyncratic solutions rather than an open standardized ecosystem. In this editorial note we argue for desirability of such an ecosystem, outline the main requirements and sketch possible solutions. We focus on enterprise as our primary use case and illustrate the main ideas through it, but the same principles apply to various different use cases.

We study the greedy-based online algorithm for edge-weighted matching with (one-sided) vertex arrivals in bipartite graphs, and edge arrivals in general graphs. This algorithm was first studied more than a decade ago by Korula and Pal for the bipartite case in the random-order model. While the weighted bipartite matching problem is solved in the random-order model, this is not the case in recent and exciting online models in which the online player is provided with a sample, and the arrival order is adversarial. The greedy-based algorithm is arguably the most natural and practical algorithm to be applied in these models. Despite its simplicity and appeal, and despite being studied in multiple works, the greedy-based algorithm was not fully understood in any of the studied online models, and its actual performance remained an open question for more than a decade. We provide a thorough analysis of the greedy-based algorithm in several online models. For vertex arrivals in bipartite graphs, we characterize the exact competitive-ratio of this algorithm in the random-order model, for any arrival order of the vertices subsequent to the sampling phase (adversarial and random orders in particular). We use it to derive tight analysis in the recent adversarial-order model with a sample (AOS model) for any sample size, providing the first result in this model beyond the simple secretary problem. Then, we generalize and strengthen the black box method of converting results in the random-order model to single-sample prophet inequalities, and use it to derive the state-of-the-art single-sample prophet inequality for the problem. Finally, we use our new techniques to analyze the greedy-based algorithm for edge arrivals in general graphs and derive results in all the mentioned online models. In this case as well, we improve upon the state-of-the-art single-sample prophet inequality.

Monitoring network traffic is an important building block for various management and security systems. Typically, the number of active flows in a network node is much larger than the number of available monitoring resources, making it imprac-tical to maintain a "per-flow" state at the node. This situation gave rise to the recent interest in streaming algorithms where complex data structures are used to perform monitoring tasks efficiently. However, these solutions often require complicated "per-packet" operations, which are not feasible in current devices line-rate. Even when the amortized (expected average) complexity is O(1) operations per-packet, some packets may experience a much longer delay, which again is impractical. In this dissertation we study three important monitoring problems (that were recently studied in the context of streaming algorithms) and for each of them we present practical, efficient resource-constrained algorithms.

The world is dynamic and changes over time, thus any optimization problem used to model real life problems must address this dynamic nature, taking into account the cost of changes to a solution over time. The multistage model was introduced with this goal in mind. In this model we are given a series of instances of an optimization problem, corresponding to different times, and a solution is provided for each instance. The strive for obtaining near-optimal solutions for each instance on one hand, while maintaining similar solutions for consecutive time units on the other hand, is quantified and integrated into the objective function. In this paper we consider the Generalized Multistage $d$-Knapsack problem, a generalization of the multistage variants of the Multiple Knapsack problem, as well as the $d$-Dimensional Knapsack problem. We present a PTAS for Generalized Multistage $d$-Knapsack.

We study the classical online bipartite matching problem: One side of the graph is known and vertices of the other side arrive online. It is well known that when the graph is edge-weighted, and vertices arrive in an adversarial order, no online algorithm has a nontrivial competitive-ratio. To bypass this hurdle we modify the rules such that the adversary still picks the graph but has to reveal a random part (say half) of it to the player. The remaining part is given to the player in an adversarial order. This models practical scenarios in which the online algorithm has some history to learn from. This way of modeling a history was formalized recently by the authors (SODA 20) and was called the AOS model (for Adversarial Online with a Sample). It allows developing online algorithms for the secretary problem that compete even when the secretaries arrive in an adversarial order. Here we use the same model to attack the much more challenging matching problem. We analyze a natural algorithmic framework that decides how to match an arriving vertex $v$ by applying an offline matching algorithm to $v$ and the sample. We get roughly $1/4$ of the maximum weight by applying the offline greedy matching algorithm to the sample and $v$. Our analysis ties the performance of this algorithm to the performance of the offline greedy matching on the online part and we also prove that it is tight. Surprisingly, when replacing greedy with an optimal algorithm for maximum matching, no constant competitive-ratio can be guaranteed when the size of the sample is comparable to the size of the online part. However, when the sample is quadratic in the size of the online part, we do get a competitive-ratio of $1/e$.

Network-wide traffic analytics are often needed for various network monitoring tasks. These measurements are often performed by collecting samples at network switches, which are then sent to the controller for aggregation. However, performing such analytics without “overcounting’' flows or packets that traverse multiple measurement switches is challenging. Therefore, existing solutions often simplify the problem by making assumptions on the routing or measurement switch placement. We introduce AROMA, a measurement infrastructure that generates a uniform sample of packets and flows regardless of the topology, workload and routing. Therefore, AROMA can be deployed in many settings, and can also work in the data plane using programmable PISA switches. The AROMA infrastructure includes controller algorithms that approximate a variety of essential measurement tasks while providing formal accuracy guarantees. Using extensive simulations on real-world network traces, we show that our algorithms are competitively accurate compared to the best existing solutions despite the fact that they make no assumptions on the underlying network or the placement of measurement switches.

We study the problem of maximizing a monotone submodular function subject to a Multiple Knapsack constraint (SMKP) . The input is a set I of items, each associated with a non-negative weight, and a set of bins, each having a capacity. Also, we are given a submodular, monotone and non-negative function f over subsets of the items. The objective is to find a subset of items A ⊆ I and a packing of the items in the bins, such that f(A) is maximized. SMKP is a natural extension of both Multiple Knapsack and the problem of monotone submodular maximization subject to a knapsack constraint. Our main result is a nearly optimal polynomial time (1−e−ε)-approximation algorithm for the problem, for any ε > 0. Our algorithm relies on a refined analysis of techniques for constrained submodular optimization combined with sophisticated application of tools used in the development of approximation schemes for packing problems.

Finding the network's heaviest flows is an important and challenging network monitoring task and a critical building block for many other applications. In the hierarchical heavy hitters (HHH) problem, one needs to identify the most frequent network IP-prefixes hierarchically. This is a challenging task since the number of relevant IP-prefixes of flows in a busy router is much higher than the number of counters. To address this point, many streaming algorithms were recently developed, but they use complex data-structures and usually have non-constant per-packet update-time, preventing them from being deployed in line-speed. A randomized constant-time algorithm was proposed recently; however, it is only applicable to extremely large streams. In this paper, we propose a constant-time algorithm for detecting the HHH that does not have any convergence requirements and achieves comparable results to state of the art. Furthermore, our algorithm uses only efficient built-in counters available in current network devices, making it deployable on commercially off-the-shelf network gear. We provide an analytical study of the problem and show, using emulation over real traffic, that our algorithm performs at least as well as the best-known streaming algorithms without performing expensive per-packet operations or requiring convergence periods.

We study the problem of maximizing a monotone submodular function subject to a Multiple Knapsack constraint (SMKP). The input is a set I of items, each associated with a non-negative weight, and a set of bins having arbitrary capacities. Also, we are given a submodular, monotone and non-negative function f over subsets of the items. The objective is to find a subset of items A ⊆ I and a packing of these items in the bins, such that f(A) is maximized. SMKP is a natural extension of both Multiple Knapsack and the problem of monotone submodular maximization subject to a knapsack constraint. Our main result is a nearly optimal polynomial time (1-e^{-1}-e)-approximation algorithm for the problem, for any e > 0. Our algorithm relies on a refined analysis of techniques for constrained submodular optimization combined with sophisticated application of tools used in the development of approximation schemes for packing problems.

The recent introduction of SDN allows deploying new centralized network algorithms that dramatically improve the network operation. Many of these solutions rely on the assumption that the centralized controller merges data from different Network Monitoring Points (NMP) to obtain a network-wide view. This is far from trivial when the same packet may traverse through several NMPs. Therefore, existing solutions either assume that each packet is measured at exactly one NMP or that the routing of each packet is known. Another approach is to mark the sampled packets so that other NMPs are aware that the packet was already considered. We suggest the first network-wide and routing oblivious algorithms for three fundamental network monitoring problems. The suggested algorithms allow flexible NMP placement, require no control over edge routers, and are indifferent to network topology and routing. Moreover, they are based on passive measurements without modifying the traffic in any way. Formally, we provide a general, constant time framework that solves the distributed versions of the volume estimation, frequency estimation and heavy-hitters problems with provable guarantees. The evaluation of our scheme on real packet traces shows that we can achieve very accurate results using a very reasonable amount of memory. For example, using less than 60KB memory in each monitoring point we get a root square error of less than 0.01% of the packets for frequency estimation.

The stochastic knapsack problem is a stochastic version of the well known deterministic knapsack problem, in which some of the input values are random variables. There are several variants of the stochastic problem. In this paper we concentrate on the chance-constrained variant, where item values are deterministic and item sizes are stochastic. The goal is to find a maximum value allocation subject to the constraint that the overflow probability is at most a given value. Previous work showed a PTAS for the problem for various distributions (Poisson, Exponential, Bernoulli and Normal). Some strictly respect the constraint and some relax the constraint by a factor of (1 + ). All algorithms use Ω(n1/ ) time. A very recent work showed a “almost FPTAS” algorithm for Bernoulli distributions with O(poly(n) · quasipoly(1/ )) time. In this paper we present a FPTAS for normal distributions with a solution that satisfies the chance constraint in a relaxed sense. The normal distribution is particularly important, because by the Berry-Esseen theorem, an algorithm solving the normal distribution also solves, under mild conditions, arbitrary independent distributions. To the best of our knowledge, this is the first (relaxed or non-relaxed) FPTAS for the problem. In fact, our algorithm runs in poly( ) time. We achieve the FPTAS by a delicate combination of previous techniques plus a new alternative solution to the non-heavy elements that is based on a non-convex program with a simple structure and an O(n2 log n ) running time. We believe this part is also interesting on its own right. 2012 ACM Subject Classification Theory of computation → Approximation algorithms analysis, Theory of computation → Stochastic approximation, Theory of computation → Discrete optimization, Theory of computation → Nonconvex optimization

Network Function Virtualization (NFV) is a novel paradigm that enables flexible and scalable implementation of network services on cloud infrastructure. A key factor in the success of NFV is the ability to dynamically allocate physical resources according to the demand. This is particularly important when dealing with the data plane since additional resources are required in order to support the virtual switching of the packets between the Virtual Network Functions (VNFs). The exact amount of these resources depends on the way service chains are deployed and the amount of network traffic being handled. Thus, orchestrating service chains that require high traffic throughput is a very complex task and most existing solutions either concentrate on handcrafted tuning of the servers to achieve the needed performance level, or present theoretical placement functions that assume that the switching cost is part of the input. In this work, we bridge this gap by presenting a deployment algorithm for service chains that optimizes performance by minimizing the actual cost of virtual switching. The results are based on extensive measurements of the actual switching cost and the performance of service chains in a realistic NFV environment. Our evaluation indicates that this new algorithm significantly reduces virtual switching resource utilization when compared to the de-facto standard placement in OpenStack/Nova - allowing a much higher acceptance ratio of network services.

Monitoring network traffic is an important building block for various management and security systems. In typical settings, the number of active flows in a network node is much larger than the number of available monitoring resources and there is no practical way to maintain per-flow state at the node. This situation gave rise to the recent interest in streaming algorithms where complex data structures are used to perform monitoring tasks like identifying the top-k flows using a constant amount of memory. However, these solutions require complicated per-packet operations, which are not feasible in current hardware or software network nodes.In this paper, we take a different approach to this problem and study the ability to perform monitoring tasks using efficient built-in counters available in current network devices. We show that by applying non-trivial control algorithms that change the filter assignments of these built-in counters at a fixed time interval, regardless of packet arrival rate, we can get accurate monitoring information. We provide an analytical study of the top-k flows problem and show, using extensive emulation over recent real traffic, that our algorithm can perform at least as well as the best-known streaming algorithms without using complex data structure or performing expensive per-packet operations.

The dynamic NFV placement problem captures one of the main challenges facing the telecom industry following the emergence of the Network Function Virtualization (NFV) networking paradigm, that is, deciding the placement of network functions while taking into consideration the dynamic nature of networks and workloads. We model the problem as a generalization of the classic Uncapacitated Facility Location (UFL) problem, where we consider both multiple types of commodities and dynamic clients whose location changes over time. We show that under reasonable assumptions we are able to develop a 7-approximation algorithm for the Dynamic Facility Location (DFL) problem, improving the logarithmic approximation of Eisenstat et al. [6]. We build upon this result to develop the first virtualized services placement algorithm that accounts for dynamic changes. Our tri-criteria approximation algorithms provide constant approximation factors with respect to the overall performance and size constraints, and logarithmic approximation factors with respect to capacity constraints.

Network Function Virtualization (NFV) is a novel paradigm that enables flexible and scalable implementation of network services on cloud infrastructure. An important enabler for the NFV paradigm is software switching, which should satisfy rigid network requirements such as high throughput and low latency. Despite recent research activities in the field of NFV, not much attention was given to understand the costs of software switching in NFV deployments. Existing approaches for traffic steering and orchestration of virtual network functions either neglect the cost of software switching or assume that it can be provided as an input, and therefore real NFV deployments of network services are often suboptimal. In this work, we conduct an extensive and in-depth evaluation that examines the impact of service chaining deployments on Open vSwitch - the de facto standard software switch for cloud environments. We provide insights on network performance metrics such as throughput, CPU utilization and packet processing, while considering different placement strategies of a service chain. We then use these insights to provide an abstract generalized cost function that accurately captures the CPU switching cost of deployed service chains. This cost is an essential building block for any practical optimized placement management and orchestration strategy for NFV service chaining.

Distributed cloud networking builds on network functions virtualization (NFV) and software defined networking (SDN) to enable the deployment of network services in the form of elastic virtual network functions (VNFs) instantiated over general purpose servers at distributed cloud locations. We address the design of fast approximation algorithms for the NFV service distribution problem (NSDP), whose goal is to determine the placement of VNFs, the routing of service flows, and the associated allocation of cloud and network resources that satisfy client demands with minimum cost. We show that in the case of load-proportional costs, the resulting fractional NSDP can be formulated as a multi-commodity-chain flow problem on a cloud-augmented graph, and design a queue-length based algorithm, named QNSD, that provides an O(ε) approximation in time O (1/ε). We then address the case in which resource costs are a function of the integer number of allocated resources and design a variation of QNSD that effectively pushes for flow consolidation into a limited number of active resources to minimize overall cloud network cost.

One of the main motivations for the shift to the Cloud (and the more recent shift of telco operators into NFV) is cost reduction due to high utilization of infrastructure resources. However, achieving high utilization in practical scenarios is complex since the term "resources" covers different orthogonal aspects, such as server CPU, storage (or disk) usage and network capacity, and the workload characterization varies over time and over different users. In this paper we study the placement of Virtual Machines (VMs) that implement services over the physical infrastructure, trying to understand what makes a placement scheme better than others in the overall utilization of the various resources. We show that the multidimensional case is inherently different from the single dimension case, and develop novel placement heuristics to address the specific challenges. We then show, by extensive evaluation over real data, that operators can significantly improve their resource utilization by selecting the most appropriate placement policy, according to their system specifications and the deployed services. In particular, two of our new heuristics that dynamically change the placement logic according to the amount of available (unused) resources are shown to perform very well in many practical scenarios.

Monitoring systems and ensuring the required service level is an important operation task. However, doing this based on external visible data, such as systems logs, is very difficult since it is very hard to extract from the logged data the exact state and the root cause to the actions taken by the system. Yet, identifying behavioral changes of complex systems can be used for early identification of problems and allow proactive correction measurements. Since it is practically impossible to perform this task manually, there is a critical need for a methodology that can analyze logs, automatically create a behavioral model, and compare the behavior to the expected behavior.In this paper we propose a novel approach for comparison between serviceexecutions as exhibited in their log files. The behavior is captured by FiniteState Automaton models (FSAs), enhanced with performance related data, bothmined from the logs. Our tool then computes the difference between the current model and behavioral models created when the service was known to operate well. A visual framework that graphically presents and emphasizes the changes in the behavior is then used to trace their root cause. We evaluate our approach over real telecommunication logs.