793 research outputs found
Faster Algorithms for Semi-Matching Problems
We consider the problem of finding \textit{semi-matching} in bipartite graphs
which is also extensively studied under various names in the scheduling
literature. We give faster algorithms for both weighted and unweighted case.
For the weighted case, we give an -time algorithm, where is
the number of vertices and is the number of edges, by exploiting the
geometric structure of the problem. This improves the classical
algorithms by Horn [Operations Research 1973] and Bruno, Coffman and Sethi
[Communications of the ACM 1974].
For the unweighted case, the bound could be improved even further. We give a
simple divide-and-conquer algorithm which runs in time,
improving two previous -time algorithms by Abraham [MSc thesis,
University of Glasgow 2003] and Harvey, Ladner, Lov\'asz and Tamir [WADS 2003
and Journal of Algorithms 2006]. We also extend this algorithm to solve the
\textit{Balance Edge Cover} problem in time, improving the
previous -time algorithm by Harada, Ono, Sadakane and Yamashita [ISAAC
2008].Comment: ICALP 201
Locally Optimal Load Balancing
This work studies distributed algorithms for locally optimal load-balancing:
We are given a graph of maximum degree , and each node has up to
units of load. The task is to distribute the load more evenly so that the loads
of adjacent nodes differ by at most .
If the graph is a path (), it is easy to solve the fractional
version of the problem in communication rounds, independently of the
number of nodes. We show that this is tight, and we show that it is possible to
solve also the discrete version of the problem in rounds in paths.
For the general case (), we show that fractional load balancing
can be solved in rounds and discrete load
balancing in rounds for some function , independently of the
number of nodes.Comment: 19 pages, 11 figure
All-Norm Load Balancing in Graph Streams via the Multiplicative Weights Update Method
In the weighted load balancing problem, the input is an n-vertex bipartite graph between a set of clients and a set of servers, and each client comes with some nonnegative real weight. The output is an assignment that maps each client to one of its adjacent servers, and the load of a server is then the sum of the weights of the clients assigned to it. The goal is to find an assignment that is well-balanced, typically captured by (approximately) minimizing either the ?_?- or ??-norm of the server loads. Generalizing both of these objectives, the all-norm load balancing problem asks for an assignment that approximately minimizes all ?_p-norm objectives for p ? 1, including p = ?, simultaneously.
Our main result is a deterministic O(log n)-pass O(1)-approximation semi-streaming algorithm for the all-norm load balancing problem. Prior to our work, only an O(log n)-pass O(log n)-approximation algorithm for the ?_?-norm objective was known in the semi-streaming setting.
Our algorithm uses a novel application of the multiplicative weights update method to a mixed covering/packing convex program for the all-norm load balancing problem involving an infinite number of constraints
Improved Bounds for Distributed Load Balancing
In the load balancing problem, the input is an -vertex bipartite graph and a positive weight for each client . The algorithm
must assign each client to an adjacent server . The load of
a server is then the weighted sum of all the clients assigned to it, and the
goal is to compute an assignment that minimizes some function of the server
loads, typically either the maximum server load (i.e., the
-norm) or the -norm of the server loads.
We study load balancing in the distributed setting. There are two existing
results in the CONGEST model. Czygrinow et al. [DISC 2012] showed a
2-approximation for unweighted clients with round-complexity ,
where is the maximum degree of the input graph. Halld\'orsson et al.
[SPAA 2015] showed an -approximation for unweighted
clients and -approximation for weighted clients
with round-complexity polylog.
In this paper, we show the first distributed algorithms to compute an
-approximation to the load balancing problem in polylog rounds. In
the CONGEST model, we give an -approximation algorithm in polylog
rounds for unweighted clients. For weighted clients, the approximation ratio is
. In the less constrained LOCAL model, we give an
-approximation algorithm for weighted clients in polylog rounds.
Our approach also has implications for the standard sequential setting in
which we obtain the first -approximation for this problem that runs in
near-linear time. A 2-approximation is already known, but it requires solving a
linear program and is hence much slower. Finally, we note that all of our
results simultaneously approximate all -norms, including the
-norm
Nearly-Linear Time LP Solvers and Rounding Algorithms for Scheduling Problems
We study nearly-linear time approximation algorithms for non-preemptive scheduling problems in two settings: the unrelated machine setting, and the identical machine with job precedence constraints setting, under the well-studied objectives such as makespan and weighted completion time. For many problems, we develop nearly-linear time approximation algorithms with approximation ratios matching the current best ones achieved in polynomial time.
Our main technique is linear programming relaxation. For the unrelated machine setting, we formulate mixed packing and covering LP relaxations of nearly-linear size, and solve them approximately using the nearly-linear time solver of Young. For the makespan objective, we develop a rounding algorithm with (2+?)-approximation ratio. For the weighted completion time objective, we prove the LP is as strong as the rectangle LP used by Im and Li, leading to a nearly-linear time (1.45 + ?)-approximation for the problem.
For problems in the identical machine with precedence constraints setting, the precedence constraints can not be formulated as packing or covering constraints. To achieve the nearly-linear running time, we define a polytope for the constraints, and leverage the multiplicative weight update (MWU) method with an oracle which always returns solutions in the polytope
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Approximation and Control of Skill Based Parallel Service Systems with Homogeneous Service
A skill base parallel service system is comprised of a set of customers of different classes that arrive randomly for service, a set of servers that serve those customers and a set of qualifications that defines which customer classes can be served by which server. Systems of this kind appear in a wide range of applications from the assignment of jobs to employees with different skills to network traffic routing. Literature regarding these systems has almost exclusively been focused on the asymptotic heavy traffic regime. The reason being that such an asymptotic regime is convenient to analyze and allows the derivation of exact results. However, although many applications can be well approximated by an asymptotic regime, many others can not. In this work we are especially concerned with large scale sparse systems where, despite the system being large of scale, each customer class can only be served by a small subset of the servers. After laying foundations for the model in Chapter 1 and exploring structural properties in Chapter 2 we go on to present the two main contributions of this work. In Chapter 3 we develop a set of approximations that compile to a , first of its kind, approximation scheme of matching rates of skill based parallel service system operating under the \textit{first-come-first-serve} or \textit{longest-queue-first} policies. The accuracy of the approximation is verified with extensive simulation experiments where it is shown to provide matching rate estimates with an absolute error of for a wide range of traffic intensities. Later, in Chapter 4 we use insights provided by the new approximation to derive weighted versions of the \textit{first-come-first-serve} or \textit{longest-queue-first} and show, through comprehensive simulation testing, that these weighted polices dramatically reduce the waiting time of customers in congested system compared to the original unweighted versions. Finally, we extend the use of the weighted policies to systems with matching rewards and show that, by appropriate choice of weights, these policies can be used by a controller to efficiently trade-off between the rate of reward accumulation and waiting time experienced by the customer
Scheduling with processing set restrictions : a survey
2008-2009 > Academic research: refereed > Publication in refereed journalAccepted ManuscriptPublishe
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