Improved Algorithms for Scheduling Unsplittable Flows on Paths

In this paper, we investigate offline and online algorithms for Round-UFPP, the problem of minimizing the number of rounds required to schedule a set of unsplittable flows of non-uniform sizes on a given path with non-uniform edge capacities. Round-UFPP is NP-hard and constant-factor approximation algorithms are known under the no bottleneck assumption (NBA), which stipulates that maximum size of a flow is at most the minimum edge capacity. We study Round-UFPP without the NBA, and present improved online and offline algorithms. We first study offline Round-UFPP for a restricted class of instances called alpha-small, where the size of each flow is at most alpha times the capacity of its bottleneck edge, and present an O(log(1/(1 - alpha)))-approximation algorithm. Our main result is an online O(log log cmax)-competitive algorithm for Round-UFPP for general instances, where cmax is the largest edge capacities, improving upon the previous best bound of O(log cmax) due to [16]. Our result leads to an offline O(min(log n, log m, log log cmax))- approximation algorithm and an online O(min(log m, log log cmax))-competitive algorithm for Round-UFPP, where n is the number of flows and m is the number of edges

A Constant Factor Approximation Algorithm for Unsplittable Flow on Paths

In the unsplittable flow problem on a path, we are given a capacitated path $P$ and $n$ tasks, each task having a demand, a profit, and start and end vertices. The goal is to compute a maximum profit set of tasks, such that for each edge $e$ of $P$, the total demand of selected tasks that use $e$ does not exceed the capacity of $e$. This is a well-studied problem that has been studied under alternative names, such as resource allocation, bandwidth allocation, resource constrained scheduling, temporal knapsack and interval packing. We present a polynomial time constant-factor approximation algorithm for this problem. This improves on the previous best known approximation ratio of $O(\log n)$. The approximation ratio of our algorithm is $7+\epsilon$ for any $\epsilon>0$. We introduce several novel algorithmic techniques, which might be of independent interest: a framework which reduces the problem to instances with a bounded range of capacities, and a new geometrically inspired dynamic program which solves a special case of the maximum weight independent set of rectangles problem to optimality. In the setting of resource augmentation, wherein the capacities can be slightly violated, we give a $(2+\epsilon)$-approximation algorithm. In addition, we show that the problem is strongly NP-hard even if all edge capacities are equal and all demands are either~1,~2, or~3.Comment: 37 pages, 5 figures Version 2 contains the same results as version 1, but the presentation has been greatly revised and improved. References have been adde

Stochastic Unsplittable Flows

We consider the stochastic unsplittable flow problem: given a graph with edge-capacities, and source-sink pairs with each pair having a size and a value, the goal is to route the pairs unsplittably while respecting edge capacities to maximize the total value of the routed pairs. However, the size of each pair is a random variable and is revealed only after we decide to route that pair. Which pairs should we route, along which paths, and in what order so as to maximize the expected value? We present results for several cases of the problem under the no-bottleneck assumption. We show a logarithmic approximation algorithm for the single-sink problem on general graphs, considerably improving on the prior results of Chawla and Roughgarden which worked for planar graphs. We present an approximation to the stochastic unsplittable flow problem on directed acyclic graphs, within less than a logarithmic factor of the best known approximation in the non-stochastic setting. We present a non-adaptive strategy on trees that is within a constant factor of the best adaptive strategy, asymptotically matching the best results for the non-stochastic unsplittable flow problem on trees. Finally, we give results for the stochastic unsplittable flow problem on general graphs. Our techniques include using edge-confluent flows for the single-sink problem in order to control the interaction between flow-paths, and a reduction from general scheduling policies to "safe" ones (i.e., those guaranteeing no capacity violations), which may be of broader interest

Survey of Consistent Network Updates

Computer networks have become a critical infrastructure. Designing dependable computer networks however is challenging, as such networks should not only meet strict requirements in terms of correctness, availability, and performance, but they should also be flexible enough to support fast updates, e.g., due to a change in the security policy, an increasing traffic demand, or a failure. The advent of Software-Defined Networks (SDNs) promises to provide such flexiblities, allowing to update networks in a fine-grained manner, also enabling a more online traffic engineering. In this paper, we present a structured survey of mechanisms and protocols to update computer networks in a fast and consistent manner. In particular, we identify and discuss the different desirable update consistency properties a network should provide, the algorithmic techniques which are needed to meet these consistency properties, their implications on the speed and costs at which updates can be performed. We also discuss the relationship of consistent network update problems to classic algorithmic optimization problems. While our survey is mainly motivated by the advent of Software-Defined Networks (SDNs), the fundamental underlying problems are not new, and we also provide a historical perspective of the subject

Time4: Time for SDN

With the rise of Software Defined Networks (SDN), there is growing interest in dynamic and centralized traffic engineering, where decisions about forwarding paths are taken dynamically from a network-wide perspective. Frequent path reconfiguration can significantly improve the network performance, but should be handled with care, so as to minimize disruptions that may occur during network updates. In this paper we introduce Time4, an approach that uses accurate time to coordinate network updates. Time4 is a powerful tool in softwarized environments, that can be used for various network update scenarios. Specifically, we characterize a set of update scenarios called flow swaps, for which Time4 is the optimal update approach, yielding less packet loss than existing update approaches. We define the lossless flow allocation problem, and formally show that in environments with frequent path allocation, scenarios that require simultaneous changes at multiple network devices are inevitable. We present the design, implementation, and evaluation of a Time4-enabled OpenFlow prototype. The prototype is publicly available as open source. Our work includes an extension to the OpenFlow protocol that has been adopted by the Open Networking Foundation (ONF), and is now included in OpenFlow 1.5. Our experimental results show the significant advantages of Time4 compared to other network update approaches, and demonstrate an SDN use case that is infeasible without Time4.Comment: This report is an extended version of "Software Defined Networks: It's About Time", which was accepted to IEEE INFOCOM 2016. A preliminary version of this report was published in arXiv in May, 201

Networks, Communication, and Computing Vol. 2

Networks, communications, and computing have become ubiquitous and inseparable parts of everyday life. This book is based on a Special Issue of the Algorithms journal, and it is devoted to the exploration of the many-faceted relationship of networks, communications, and computing. The included papers explore the current state-of-the-art research in these areas, with a particular interest in the interactions among the fields

Approximating the single source unsplittable min-cost flow problem

In the single source unsplittable min-cost flow problem, commodities must be routed simultaneously from a common source vertex to certain destination vertices in a given graph with edge capacities and costs; the demand of each commodity must be routed along a single path and the total cost must not exceed a given budget. This problem has been introduced by Kleinberg and generalizes several NP-complete problems from various areas in combinatorial optimization such as packing, partitioning, scheduling, load balancing and virtual-circuit routing