213 research outputs found
Ranking with Submodular Valuations
We study the problem of ranking with submodular valuations. An instance of
this problem consists of a ground set , and a collection of monotone
submodular set functions , where each .
An additional ingredient of the input is a weight vector . The
objective is to find a linear ordering of the ground set elements that
minimizes the weighted cover time of the functions. The cover time of a
function is the minimal number of elements in the prefix of the linear ordering
that form a set whose corresponding function value is greater than a unit
threshold value.
Our main contribution is an -approximation algorithm
for the problem, where is the smallest non-zero marginal value that
any function may gain from some element. Our algorithm orders the elements
using an adaptive residual updates scheme, which may be of independent
interest. We also prove that the problem is -hard to
approximate, unless P = NP. This implies that the outcome of our algorithm is
optimal up to constant factors.Comment: 16 pages, 3 figure
Improved Online Algorithm for Weighted Flow Time
We discuss one of the most fundamental scheduling problem of processing jobs
on a single machine to minimize the weighted flow time (weighted response
time). Our main result is a -competitive algorithm, where is the
maximum-to-minimum processing time ratio, improving upon the
-competitive algorithm of Chekuri, Khanna and Zhu (STOC 2001). We
also design a -competitive algorithm, where is the
maximum-to-minimum density ratio of jobs. Finally, we show how to combine these
results with the result of Bansal and Dhamdhere (SODA 2003) to achieve a
-competitive algorithm (where is the
maximum-to-minimum weight ratio), without knowing in advance. As shown
by Bansal and Chan (SODA 2009), no constant-competitive algorithm is achievable
for this problem.Comment: 20 pages, 4 figure
Admission Control to Minimize Rejections and Online Set Cover with Repetitions
We study the admission control problem in general networks. Communication
requests arrive over time, and the online algorithm accepts or rejects each
request while maintaining the capacity limitations of the network. The
admission control problem has been usually analyzed as a benefit problem, where
the goal is to devise an online algorithm that accepts the maximum number of
requests possible. The problem with this objective function is that even
algorithms with optimal competitive ratios may reject almost all of the
requests, when it would have been possible to reject only a few. This could be
inappropriate for settings in which rejections are intended to be rare events.
In this paper, we consider preemptive online algorithms whose goal is to
minimize the number of rejected requests. Each request arrives together with
the path it should be routed on. We show an -competitive
randomized algorithm for the weighted case, where is the number of edges in
the graph and is the maximum edge capacity. For the unweighted case, we
give an -competitive randomized algorithm. This settles an
open question of Blum, Kalai and Kleinberg raised in \cite{BlKaKl01}. We note
that allowing preemption and handling requests with given paths are essential
for avoiding trivial lower bounds
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