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A constant factor approximation algorithm for the storage allocation problem: extended abstract
 In SPAA
, 2013
"... We study the STORAGE ALLOCATION PROBLEM (SAP) which is a variant of the UNSPLITTABLE FLOW PROBLEM ON PATHS (UFPP). A SAP instance consists of a path P = (V,E) and a set J of tasks. Each edge e ∈ E has a capacity ce and each task j ∈ J is associated with a path Ij in P, a demand dj and a weight wj. T ..."
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We study the STORAGE ALLOCATION PROBLEM (SAP) which is a variant of the UNSPLITTABLE FLOW PROBLEM ON PATHS (UFPP). A SAP instance consists of a path P = (V,E) and a set J of tasks. Each edge e ∈ E has a capacity ce and each task j ∈ J is associated with a path Ij in P, a demand dj and a weight wj. The goal is to find a maximum weight subset S ⊆ J of tasks and a height function h: S → R+ such that (i) h(j) + dj ≤ ce, for every e ∈ Ij; and (ii) if j, i ∈ S such that Ij ∩ Ii 6 = ∅ and h(j) ≥ h(i), then h(j) ≥ h(i) + di. SAP can be seen as a rectangle packing problem in which rectangles can be moved vertically, but not horizontally. We present a polynomial time (9 + ε)approximation algorithm for SAP. Our algorithm is based on a variation of the framework for approximating UFPP by Bonsma et al. [FOCS 2011] and on a (4 + ε)approximation algorithm for δsmall SAP instances, namely for instances in which dj ≤ δ · ce, for every e ∈ Ij, for a sufficiently small constant δ> 0. In our algorithm for δsmall instances, tasks are packed carefully in strips in a UFPP manner, and then a (1 + ε) factor is incurred by a reduction from SAP to UFPP in strips. The strips are stacked to form a SAP solution. Finally, we show that SAP is strongly NPhard, even with uniform weights and even if assuming the no bottleneck assumption.
Improved Approximation Algorithms for Unsplittable Flow on a Path with Time Windows
"... In the wellstudied Unsplittable Flow on a Path problem (UFP), we are given a path graph with edge capacities. Furthermore, we are given a collection of n tasks, each one characterized by a subpath, a weight, and a demand. Our goal is to select a maximum weight subset of tasks so that the total de ..."
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In the wellstudied Unsplittable Flow on a Path problem (UFP), we are given a path graph with edge capacities. Furthermore, we are given a collection of n tasks, each one characterized by a subpath, a weight, and a demand. Our goal is to select a maximum weight subset of tasks so that the total demand of selected tasks using each edge is upper bounded by the corresponding capacity. Chakaravarthy et al. [ESA’14] studied a generalization of UFP, bagUFP, where tasks are partitioned into bags, and we can select at most one task per bag. Intuitively, bags model jobs that can be executed at different times (with different duration, weight, and demand). They gave a O(logn) approximation for bagUFP. This is also the best known ratio in the case of uniform weights. In this paper we achieve the following main results: •We present an LPbased O(logn / log logn) approximation for bagUFP. We remark that, prior to our work, the best known integrality gap (for a nonextended formulation) was O(logn) even in the special case of UFP [Chekuri et al., APPROX’09]. •We present an LPbasedO(1) approximation for uniformweight bagUFP. This also generalizes the integrality gap bound for uniformweight UFP by Anagnostopoulos et al. [IPCO’13]. •We consider a relevant special case of bagUFP, twUFP, where tasks in a bag model the possible ways in which we can schedule a job with a given processing time within a given time window. We present a QPTAS for twUFP with quasipolynomial demands and under the Bounded TimeWindow Assumption, i.e. assuming that the time window size of each job is within a constant factor from its processing time. This generalizes the QPTAS for UFP by Bansal et al. [STOC’06].
Approximation Algorithms for the Unsplittable Flow Problem on Paths and Trees
"... We study the Unsplittable Flow Problem (UFP) and related variants, namely UFP with Bag Constraints and UFP with Rounds, on paths and trees. We provide improved constant factor approximation algorithms for all these problems under the no bottleneck assumption (NBA), which says that the maximum demand ..."
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We study the Unsplittable Flow Problem (UFP) and related variants, namely UFP with Bag Constraints and UFP with Rounds, on paths and trees. We provide improved constant factor approximation algorithms for all these problems under the no bottleneck assumption (NBA), which says that the maximum demand for any sourcesink pair is at most the minimum capacity of any edge. We obtain these improved results by expressing a feasible solution to a natural LP relaxation of the UFP as a nearconvex combination of feasible integral solutions.
Protecting Virtual Networks with DRONE
"... Abstract—Network virtualization is enabling infrastructure providers (InPs) to offer new services to higher level service providers (SPs). InPs are usually bound by Service Level Agreements (SLAs) to ensure various levels of resource availability for different SPs ’ virtual networks (VNs). They pro ..."
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Abstract—Network virtualization is enabling infrastructure providers (InPs) to offer new services to higher level service providers (SPs). InPs are usually bound by Service Level Agreements (SLAs) to ensure various levels of resource availability for different SPs ’ virtual networks (VNs). They provision redundant backup resources while embedding an SP’s VN request to conform to the SLAs during physical failures in the infrastructure. An extreme of this backup resource provisioning is to reserve a mutually exclusive backup of each element in an SP’s VN request. Such dedicated protection scheme can enable an InP to ensure fast VN recovery, thus, providing high uptime guarantee to the SPs. In this paper, we study the 1 + 1Protected Virtual Network Embedding (1 + 1ProViNE) problem. We propose Dedicated Protection for Virtual Network Embedding (DRONE), a suite of solutions to the 1 + 1ProViNE. DRONE includes an Integer Linear Programming (ILP) formulation for optimal solution (OPTDRONE) and a heuristic (FASTDRONE) to tackle the computational complexity in computing the optimal solution. Trace driven simulations show that FASTDRONE allocates only 14.3 % extra backup resources on average compared to the optimal solution, while executing 200x – 12000x faster. I.