Liane Lewin-Eytan

Near optimal placement of virtual network functions

Network Function Virtualization (NFV) is a new networking paradigm where network functions are executed on commodity servers located in small cloud nodes distributed across the network, and where software defined mechanisms are used to control the network flows. This paradigm is a major turning point in the evolution of networking, as it introduces high expectations for enhanced economical network services, as well as major technical challenges. In this paper, we address one of the main technical challenges in this domain: the actual placement of the virtual functions within the physical network. This placement has a critical impact on the performance of the network, as well as on its reliability and operation cost. We perform a thorough study of the NFV location problem, show that it introduces a new type of optimization problems, and provide near optimal approximation algorithms guaranteeing a placement with theoretically proven performance. The performance of the solution is evaluated with respect to two measures: the distance cost between the clients and the virtual functions by which they are served, as well as the setup costs of these functions. We provide bi-criteria solutions reaching constant approximation factors with respect to the overall performance, and adhering to the capacity constraints of the networking infrastructure by a constant factor as well. Finally, using extensive simulations, we show that the proposed algorithms perform well in many realistic scenarios.

Non-cooperative multicast and facility location games

We consider a multicast game with selfish non- cooperative players. There is a special source node and each player is interested in connecting to the source by making a routing decision that minimizes its payment. The mutual influence of the players is determined by a cost sharing mechanism, which in our case evenly splits the cost of an edge among the players using it. We consider two different models: an integral model, where each player connects to the source by choosing a single path, and a fractional model, where a player is allowed to split the flow it receives from the source between several paths. In both models we explore the overhead incurred in network cost due to the selfish behavior of the users, as well as the computational complexity of finding a Nash equilibrium. The existence of a Nash equilibrium for the integral model was previously established by the means of a potential function. We prove that finding a Nash equilibrium that minimizes the potential function is NP-hard. We focus on the price of anarchy of a Nash equilibrium resulting from the best-response dynamics of a game course, where the players join the game sequentially. For a game with in players, we establish an upper bound of O(radicnlog2 n) on the price of anarchy, and a lower bound of Omega(log n/log log n). For the fractional model, we prove the existence of a Nash equilibrium via a potential function and give a polynomial time algorithm for computing an equilibrium that minimizes the potential function. Finally, we consider a weighted extension of the multicast game, and prove that in the fractional model, the game always has a Nash equilibrium.

Non-Cooperative Multicast and Facility Location Games

We consider a multicast game with selfish non- cooperative players. There is a special source node and each player is interested in connecting to the source by making a routing decision that minimizes its payment. The mutual influence of the players is determined by a cost sharing mechanism, which in our case evenly splits the cost of an edge among the players using it. We consider two different models: an integral model, where each player connects to the source by choosing a single path, and a fractional model, where a player is allowed to split the flow it receives from the source between several paths. In both models we explore the overhead incurred in network cost due to the selfish behavior of the users, as well as the computational complexity of finding a Nash equilibrium. The existence of a Nash equilibrium for the integral model was previously established by the means of a potential function. We prove that finding a Nash equilibrium that minimizes the potential function is NP-hard. We focus on the price of anarchy of a Nash equilibrium resulting from the best-response dynamics of a game course, where the players join the game sequentially. For a game with in players, we establish an upper bound of O(radicnlog2 n) on the price of anarchy, and a lower bound of Omega(log n/log log n). For the fractional model, we prove the existence of a Nash equilibrium via a potential function and give a polynomial time algorithm for computing an equilibrium that minimizes the potential function. Finally, we consider a weighted extension of the multicast game, and prove that in the fractional model, the game always has a Nash equilibrium.

Almost optimal virtual machine placement for traffic intense data centers

The recent growing popularity of cloud-based solutions and the variety of new applications present new challenges for cloud management and resource utilization. In this paper we concentrate on the networking aspect and consider the placement problem of virtual machines (VMs) of applications with intense bandwidth requirements. Optimizing the available network bandwidth is far more complex than optimizing resources like memory or CPU, since every network link may be used by many physical hosts and thus by the VMs residing in these hosts. We focus on maximizing the benefit from the overall communication sent by the VMs to a single designated point in the data center (called the root). This is the typical case when considering a storage area network of applications with intense storage requirements. We formulate a bandwidth-constrained VM placement optimization problem that models this setting. This problem is NP hard, and we present a polynomial-time constant approximation algorithm for its most general version, in which hosts are connected to the root by a general network graph. For more practical cases, in which the network topology is a tree and the revenue is a simple function of the allocated bandwidth, we present improved approximation algorithms that are more efficient in terms of running time. We evaluate the expected performance of our proposed algorithms through a simulation study over traces from a real production data center, providing strong indications to the superiority of our proposed solutions.

Routing and Admission Control in Networks with Advance Reservations

The provisioning of quality-of-service (QoS) for real-time network applications may require the network to reserve resources. A natural way to do this is to allow advance reservations of network resources prior to the time they are needed. We consider several two-dimensional admission control problems in simple topologies such as a line, a ring and a tree. The input is a set of connection requests, each specifying its spatial characteristics, that is, its source and destination; its temporal characteristics, that is, its start time and duration time; and, potentially, also a bandwidth requirement. In addition, each request has a profit gained by acommodating it. We address the related admission control problem, where the goal is to maximize the total profit gained by the accommodated requests. We provide approximation algorithms for several problem variations. Our results imply a 4c-approximation algorithm for finding a maximum weight independent set of axis-parallel rectangles in the plane, where c is the size of a maximum set of overlapping requests.

Hedonic clustering games

Clustering, the partitioning of objects with respect to a similarity measure, has been extensively studied as a global optimization problem. We investigate clustering from a game theoretic approach, and consider the class of hedonic clustering games. Here, a self organized clustering is obtained via decisions made by independent players, corresponding to the elements clustered. Being a hedonic setting, the utility of each player is determined by the identity of the other members of her cluster. This class of games seems to be quite robust, as it fits with rather different, yet commonly used, clustering criteria. Specifically, we investigate hedonic clustering games in two different models: fixed clustering, which subdivides into k-median and k-center, and correlation clustering. We provide a thorough and non-trivial analysis of these games, characterizing Nash equilibria, and proving upper and lower bounds on the price of anarchy and price of stability. For fixed clustering we focus on the existence of a Nash equilibrium, as it is a rather non-trivial issue in this setting. We study it both for general metrics and special cases, such as line and tree metrics. In the correlation clustering model, we study both minimization and maximization variants, and provide almost tight bounds on both price of anarchy and price of stability.

Admission Control in Networks with Advance Reservations

The provisioning of quality-of-service for real-time network applications may require the network to reserve resources. A natural way to do this is to allow advance reservations of network resources prior to the time they are needed. We consider several two-dimensional admission control problems in simple topologies such as a line and a tree. The input is a set of connection requests, each specifying its spatial characteristics, that is, its source and destination; its temporal characteristics, that is, its start time and duration time; and, potentially, also a bandwidth requirement. In addition, each request is associated with a profit gained by accommodating it. We address the related admission control problem, where the goal is to maximize the total profit gained by the accommodated requests. We provide approximation algorithms for several problem variations. Our results imply a 4c-approximation algorithm for finding a maximum weight independent set of axis-parallel rectangles in the plane, where c is the size of a maximum set of overlapping rectangles.

Data-quality-aware volume reduction in smart grid networks

The power industry is in the early stages of a fundamental change, driving the integration of energy, communications, and information technologies into one intelligent utility network, known as the smart grid. This work investigates data traffic management in smart grid networks, in which huge volumes of data produced by advanced meters cannot be fully delivered to utility data centers due to limited bandwidth. To develop a solution for optimizing traffic flow, we exploit a particular characteristic of this network - power-related applications can benefit from different levels of data quality along the path to the final destination. We thus handle congestion by performing intelligent quality-aware volume reduction of the flows within the network. Our optimization problem is that of computing for each flow the amount of volume reductions in different locations, so as to maximize overall revenue. This work initiates a rigorous treatment of traffic management in the smart grid.

Dynamic Power Allocation Under Arbitrary Varying Channels - An Online Approach

A major problem in wireless networks is coping with limited resources, such as bandwidth and energy. These issues become a major algorithmic challenge in view of the dynamic nature of the wireless domain. We consider in this paper the single-transmitter power assignment problem under time-varying channels, with the objective of maximizing the data throughput. It is assumed that the transmitter has a limited power budget, to be sequentially divided during the lifetime of the battery. We deviate from the classic work in this area, which leads to explicit “water-filling” solutions, by considering a realistic scenario where the channel state quality changes arbitrarily from one transmission to the other. The problem is accordingly tackled within the framework of competitive analysis, which allows for worst-case performance guarantees in setups with arbitrarily varying channel conditions. We address both a “discrete” case, where the transmitter can transmit only at a fixed power level, and a “continuous” case, where the transmitter can choose any power level out of a bounded interval. For both cases, we propose online power-allocation algorithms with proven worst-case performance bounds. In addition, we establish lower bounds on the worst-case performance of any online algorithm and show that our proposed algorithms are optimal.

Dynamic Power Allocation Under Arbitrary Varying Channels - The Multi-User Case

We consider the power control problem in a time-slotted wireless channel, shared by a finite number of mobiles that transmit to a common base station. The channel between each mobile and the base station is time varying, and the system objective is to maximize the overall data throughput. It is assumed that each transmitter has a limited power budget, to be sequentially divided during the lifetime of the battery. We deviate from the classic work in this area, by considering a realistic scenario where the channel quality of each mobile changes arbitrarily from one transmission to the other. Assuming first that each mobile is aware of the channel quality of all other mobiles, we propose an online power-allocation algorithm, and prove its optimality under mild assumptions. We then indicate how to implement the algorithm when only local state information is available, requiring minimal communication overhead. Notably, the competitive ratio of our algorithm (nearly) matches the one we previously obtained for the (much simpler) single-transmitter case [BLMNO09], albeit requiring significantly different algorithmic solutions.