Cliff Stein

LP decoding corrects a constant fraction of errors

We show that for low-density parity-check (LDPC) codes with sufficient expansion, the linear programming (LP) decoder corrects a constant fraction of errors.

Minimizing the Total Weighted Completion Time of Coflows in Datacenter Networks

Communications in datacenter jobs (such as the shuffle operations in MapReduce applications) often involve many parallel flows, which may be processed simultaneously. This highly parallel structure presents new scheduling challenges in optimizing job-level performance objectives in data centers. Chowdhury and Stoica introduced the coflow abstraction to capture these communication patterns, and recently Chowdhury et al. developed effective heuristics to schedule coflows. In this paper, we consider the problem of efficiently scheduling coflows with release dates so as to minimize the total weighted completion time, which has been shown to be strongly NP-hard. Our main result is the first polynomial-time deterministic approximation algorithm for this problem, with an approximation ratio of 67/3, and a randomized version of the algorithm, with a ratio of 9+16√2/3. Our results use techniques from both combinatorial scheduling and matching theory, and rely on a clever grouping of coflows. We also run experiments on a Facebook trace to test the practical performance of several algorithms, including our deterministic algorithm. Our experiments suggest that simple algorithms provide effective approximations of the optimal, and that our deterministic algorithm has near-optimal performance.

Non-Preemptive Min-Sum Scheduling with Resource Augmentation

We give the first O(l)-speed O(l) approximation polynomial-time algorithms for several nonpreemptive min-sum scheduling problems where jobs arrive over time and must be processed on one machine. More precisely, we give the first O(l)-speed O(l)-approximations for the non-preemptive scheduling problems; l|r_j| Sigmaw_jF_j (weighted flow time), l |r_j| SigmaT_j (total tardiness), the broadcast version of 1 |r_j| Sigmaw_jF_j , an O(I)-speed, 1-approximation for l |r_j| Sigma U macr_j (throughput maximization), and an O(l)-machine, O(l)-speed O(1)-approximation for l |r_j| Sigmaw_jT_j (weighted tardiness). Our main contribution is an integer programming formulation whose relaxation is sufficiently close to the integer optimum, and which can be transformed to a schedule on a faster machine.

Multicast routing for energy minimization using speed scaling

We consider virtual circuit multicast routing in a network of links that are speed scalable. We assume that a link with load f uses power σ+fα, where σ is the static power, and α>1 is some constant. We assume that a link may be shutdown if not in use. In response to the arrival of client i at vertex ti a routing path (the virtual circuit) Pi connecting a fixed source s to sink ti must be established. The objective is to minimize the aggregate power used by all links. We give a polylog-competitive online algorithm, and a polynomial-time O(α)-approximation offline algorithm if the power functions of all links are the same. If each link can have a different power function, we show that the problem is APX-hard. If additionally, the edges may be directed, then we show that no poly-log approximation is possible in polynomial time under standard complexity assumptions. These are the first results on multicast routing in speed scalable networks in the algorithmic literature.

Cluster before you hallucinate: approximating node-capacitated network design and energy efficient routing

We consider circuit routing with an objective of minimizing energy, in a network of routers that are speed scalable and that may be shutdown when idle. It is known that this energy minimization problem can be reduced to a capacitated flow network design problem, where vertices have a common capacity but arbitrary costs, and the goal is to choose a minimum cost collection of vertices whose induced subgraph will support the specified flow requirements. For the multicast (single-sink) capacitated design problem we give a polynomial-time algorithm that is O(log3 n)- approximate with O(log4 n) congestion. This translates back to a O(log4α+3 n)-approximation for the multicast energy-minimization routing problem, where α is the polynomial exponent in the dynamic power used by a router. For the unicast (multicommodity) capacitated design problem we give a polynomial-time algorithm that is O(log5 n)-approximate with O(log12 n) congestion, which translates back to a O(log12α+5 n)-approximation for the unicast energy-minimization routing problem.

Single machine scheduling with job-dependent convex cost and arbitrary precedence constraints

Abstract In this work we combine resource augmentation and alpha-point scheduling techniques, which have resulted in very good performance scheduling algorithms, to compute approximate solutions for a general family of scheduling problems: each job has a convex non-decreasing cost function and the goal is to compute a schedule that minimizes the total cost subject to precedence constraints. We show that our algorithm is a O ( 1 ) -speed 1 -approximation algorithm and our numerical experiments show that the speed-scaling ratio required is close to 1 .

Divide-and-Conquer Approximation Algorithm for Vertex Cover

The vertex cover problem is a classical NP-complete problem for which the best worst-case approximation ratio is

A family of latent variable convex relaxations for IBM model 2

2-o(1)

The complexity of scheduling for p-norms of flow and stretch

. In this paper, we use a collection of simple graph transformations, each of which guarantees an approximation ratio of

An Empirical Study of Online Packet Scheduling Algorithms

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