Online Mixed Packing and Covering

Bhaskar, Umang; Fleischer, Lisa

doi:10.1137/1.9781611973105.6

Cited by 36 publications

(81 citation statements)

References 45 publications

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“…For the applications of their general framework in solving numerous online problems, we refer the reader to the survey in [11]. Azar et al [8] generalize this method for the fractional mixed packing and covering LPs. In particular, they show an application of their method for integrally solving a generalization of capacitated set cover.…”

Section: Related Online Problemsmentioning

confidence: 99%

Online Degree-Bounded Steiner Network Design

Sina¹,

Ehsani²,

Hajiaghayi³

et al. 2015

Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms

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We initiate the study of degree-bounded network design problems in the online setting. The degree-bounded Steiner tree problem -which asks for a subgraph with minimum degree that connects a given set of vertices -is perhaps one of the most representative problems in this class. This paper deals with its well-studied generalization called the degree-bounded Steiner forest problem where the connectivity demands are represented by vertex pairs that need to be individually connected. In the classical online model, the input graph is given offline but the demand pairs arrive sequentially in online steps. The selected subgraph starts off as the empty subgraph, but has to be augmented to satisfy the new connectivity constraint in each online step. The goal is to be competitive against an adversary that knows the input in advance.We design a simple greedy-like algorithm that achieves a competitive ratio of O(log n) where n is the number of vertices. We show that no (randomized) algorithm can achieve a (multiplicative) competitive ratio o(log n); thus our result is asymptotically tight. We further show strong hardness results for the group Steiner tree and the edge-weighted variants of degree-bounded connectivity problems.Fürer and Raghavachari resolved the offline variant of degree-bounded Steiner forest in their paper in SODA'92. Since then, the natural family of degree-bounded network design problems has been extensively studied in the literature resulting in the development of many interesting tools and numerous papers on the topic. We hope that our approach in this paper, paves the way for solving the online variants of the classical problems in this family of network design problems.

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Section: Related Online Problemsmentioning

confidence: 99%

Online Degree-Bounded Steiner Network Design

Sina¹,

Ehsani²,

Hajiaghayi³

et al. 2015

Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms

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“…Online algorithms for the fractional mixed packing and covering problem are investigated by Azar et al [1], where the packing constraints are known offline, while the covering constraints get revealed online. Their objective is to minimize the maximum multiplicative factor by which any constraint is getting violated, while meeting all the covering constraints.…”

Section: B Related Workmentioning

confidence: 99%

“…It can be seen that the isolation cost of both tasks is 1. Any integral allocation would have to allocate the high cost resource to one of the tasks, which results in a γ = L. However, the above linear program would return a feasible solution for γ = 1, by allocating fraction1 2 of resource 1 to both tasks.…”

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confidence: 99%

Fair Resource Allocation for Heterogeneous Tasks

Mukherjee¹,

Dutta²,

Raravi³

et al. 2015

2015 IEEE International Parallel and Distributed Processing Symposium

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We consider the problem of fair resource allocation for tasks where a resource can be assigned to at most one task, without any fractional allocation. The system is heterogeneous: capacity and cost may vary across resources, and different tasks may have different resource demand. Due to heterogeneity of resources, the cost of allocating a task in isolation, without any other competing task, may differ significantly from its allocation cost when the task is allocated along with other tasks. In this context, we consider the problem of allocating resource to tasks, while ensuring that the cost is distributed fairly across the tasks, namely, the ratio of allocation cost of a task to its isolation cost is minimized over all tasks. We show that this fair resource allocation problem is strongly NP-Hard even when the resources are of unit size by a reduction from 3-partition. Our central results are an LP rounding based algorithm with an approximation ratio of 2 + O( ) for the problem when resources are of unit size, and a near-optimal greedy algorithm for a more restricted version.The above fair allocation problem arises for resource allocation in various context, such as, allocating computing resources for reservations requests from tenants in a data center, allocating resources to computing tasks in grid computing, or allocating personnel for tasks in service delivery organizations.

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“…Linear (or convex) programs and dual fitting approaches have been popular for online scheduling; for an overview of online scheduling see [32]. Though [6] study a general online packing and covering framework, it does not capture temporal aspects of scheduling and is very different from our framework. Our work is also different from [6] from the technical point of view.…”

Section: Related Workmentioning

confidence: 99%

“…Though [6] study a general online packing and covering framework, it does not capture temporal aspects of scheduling and is very different from our framework. Our work is also different from [6] from the technical point of view. Our algorithm uses a natural algorithm PF and dual fitting using KKT conditions while [6] uses the multiplicative weights update method.…”

Section: Related Workmentioning

confidence: 99%

Competitive algorithms from competitive equilibria

Kulkarni

Munagala

2014

Proceedings of the Forty-Sixth Annual ACM Symposium on Theory of Computing

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We introduce and study a general scheduling problem that we term the Packing Scheduling problem (PSP). In this problem, jobs can have different arrival times and sizes; a scheduler can process job j at rate xj, subject to arbitrary packing constraints over the set of rates (x) of the outstanding jobs. The PSP framework captures a variety of scheduling problems, including the classical problems of unrelated machines scheduling, broadcast scheduling, and scheduling jobs of different parallelizability. It also captures scheduling constraints arising in diverse modern environments ranging from individual computer architectures to data centers. More concretely, PSP models multidimensional resource requirements and parallelizability, as well as network bandwidth requirements found in data center scheduling.In this paper, we design non-clairvoyant online algorithms for PSP and its special cases -in this setting, the scheduler is unaware of the sizes of jobs. Our results are summarized as follows.• For minimizing total weighted completion time, we show a O(1)-competitive algorithm. Surprisingly, we achieve this result by applying the well-known Proportional Fairness algorithm (PF) to perform allocations each time instant. Though PF has been extensively studied in the context of maximizing fairness in resource allocation, we present the first analysis in adversarial and general settings for optimizing job latency. Our result is also the first O(1)-competitive algorithm for weighted completion time for several classical nonclairvoyant scheduling problems. • For minimizing total weighted flow time, for any constant > 0, any O(n 1− )-competitive algorithm requires extra speed (resource augmentation) compared to the offline optimum. We show that PF is a O(log n)- * speed O(log n)-competitive non-clairvoyant algorithm, where n is the total number of jobs. We further show that there is an instance of PSP for which no nonclairvoyant algorithm can be O(n 1− )-competitive with o( √ log n) speed. • For the classical problem of minimizing total flow time for unrelated machines in the non-clairvoyant setting, we present the first online algorithm which is scalable ((1 + )-speed O(1)-competitive for any constant > 0). No non-trivial results were known for this setting, and the previous scalable algorithm could handle only related machines. We develop new algorithmic techniques to handle the unrelated machines setting that build on a new single machine scheduling policy. Since unrelated machine scheduling is a special case of PSP, when contrasted with the lower bound for PSP, our result also shows that PSP is significantly harder than perhaps the most general classical scheduling settings.Our results for PSP show that instantaneous fair scheduling algorithms can also be effective tools for minimizing the overall job latency, even when the scheduling decisions are non-clairvoyant and constrained by general packing constraints.

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Online Mixed Packing and Covering

Cited by 36 publications

References 45 publications

Online Degree-Bounded Steiner Network Design

Online Degree-Bounded Steiner Network Design

Fair Resource Allocation for Heterogeneous Tasks

Competitive algorithms from competitive equilibria

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