179 research outputs found
Simultaneously Load Balancing for Every p-norm, With Reassignments
This paper investigates the task of load balancing where the objective function is to minimize the p-norm of loads, for pgeq 1, in both static and incremental settings. We consider two closely related load balancing problems. In the bipartite matching problem we are given a bipartite graph G=(Ccup S, E) and the goal is to assign each client cin C to a server sin S so that the p-norm of assignment loads on S is minimized.
In the graph orientation problem the goal is to orient (direct) the edges of a given undirected graph while minimizing the p-norm of the out-degrees. The graph orientation problem is a special case of the bipartite matching problem, but less complex, which leads to simpler algorithms.
For the graph orientation problem we show that the celebrated Chiba-Nishizeki peeling algorithm provides a simple linear time load balancing scheme whose output is an orientation that is 2-competitive, in a p-norm sense, for all pgeq 1. For the bipartite matching problem we first provide an offline algorithm that computes an optimal assignment. We then extend this solution to the online bipartite matching problem with reassignments, where vertices from C arrive in an online fashion together with their corresponding edges, and we are allowed to reassign an amortized O(1) vertices from C each time a new vertex arrives. In this online scenario we show how to maintain a single assignment that is 8-competitive, in a p-norm sense, for all pgeq 1
Online load balancing with general reassignment cost
We investigate a semi-online variant of load balancing with restricted assignment. In this problem, we are given n jobs, which need to be processed by m machines with the goal to minimize the maximum machine load. Since strong lower bounds rule out any competitive ratio of o(logn), we may reassign jobs at a certain job-individual cost. We generalize a result by Gupta, Kumar, and Stein (SODA 2014) by giving a O(loglogmn)-competitive algorithm with constant amortized reassignment cost
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
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
SELFISHMIGRATE: A Scalable Algorithm for Non-clairvoyantly Scheduling Heterogeneous Processors
We consider the classical problem of minimizing the total weighted flow-time
for unrelated machines in the online \emph{non-clairvoyant} setting. In this
problem, a set of jobs arrive over time to be scheduled on a set of
machines. Each job has processing length , weight , and is
processed at a rate of when scheduled on machine . The online
scheduler knows the values of and upon arrival of the job,
but is not aware of the quantity . We present the {\em first} online
algorithm that is {\em scalable} ((1+\eps)-speed
-competitive for any constant \eps > 0) for the
total weighted flow-time objective. No non-trivial results were known for this
setting, except for the most basic case of identical machines. Our result
resolves a major open problem in online scheduling theory. Moreover, we also
show that no job needs more than a logarithmic number of migrations. We further
extend our result and give a scalable algorithm for the objective of minimizing
total weighted flow-time plus energy cost for the case of unrelated machines
and obtain a scalable algorithm. The key algorithmic idea is to let jobs
migrate selfishly until they converge to an equilibrium. Towards this end, we
define a game where each job's utility which is closely tied to the
instantaneous increase in the objective the job is responsible for, and each
machine declares a policy that assigns priorities to jobs based on when they
migrate to it, and the execution speeds. This has a spirit similar to
coordination mechanisms that attempt to achieve near optimum welfare in the
presence of selfish agents (jobs). To the best our knowledge, this is the first
work that demonstrates the usefulness of ideas from coordination mechanisms and
Nash equilibria for designing and analyzing online algorithms
A Robust PTAS for Machine Covering and Packing
Scheduling a set of n jobs on m identical parallel machines so as to minimize the makespan or maximize the minimum machine load are two of the most important and fundamental scheduling problems studied in the literature. We consider the general online scenario where jobs are consecutively added to and/or deleted from an instance. The goal is to always maintain a (close to) optimal assignment of the current set of jobs to the m machines. This goal is essentially doomed to failure unless, upon arrival or departure of a job, we allow to reassign some other jobs. Considering that the reassignment of a job induces a cost proportional to its size, the total cost for reassigning jobs must preferably be bounded by a constant r times the total size of added or deleted jobs
Sharing Public Land Decision Making: The Quincy Library Group Experience [includes first three items from Appendix A]
25 pages (includes illustrations).
Contains 1 reference.
Includes first three items from Appendix A
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