235,238 research outputs found
Online Set Cover with Set Requests
We consider a generic online allocation problem that generalizes the classical online set cover framework by considering requests comprising a set of elements rather than a single element. This problem has multiple applications in cloud computing, crowd sourcing, facility planning, etc. Formally, it is an online covering problem where each online step comprises an offline covering problem. In addition, the covering sets are capacitated, leading to packing constraints. We give a randomized algorithm for this problem that has a nearly tight competitive ratio in both objectives: overall cost and maximum capacity violation. Our main technical tool is an online algorithm for packing/covering LPs with nested constraints, which may be of interest in other applications as well
Online Set Cover with Set Requests *
Abstract We consider a generic online allocation problem that generalizes the classical online set cover framework by considering requests comprising a set of elements rather than a single element. This problem has multiple applications in cloud computing, crowd sourcing, facility planning, etc. Formally, it is an online covering problem where each online step comprises an offline covering problem. In addition, the covering sets are capacitated, leading to packing constraints. We give a randomized algorithm for this problem that has a nearly tight competitive ratio in both objectives: overall cost and maximum capacity violation. Our main technical tool is an online algorithm for packing/covering LPs with nested constraints, which may be of interest in other applications as well
Set Cover with Delay - Clairvoyance Is Not Required
In most online problems with delay, clairvoyance (i.e. knowing the future delay of a request upon its arrival) is required for polylogarithmic competitiveness. In this paper, we show that this is not the case for set cover with delay (SCD) - specifically, we present the first non-clairvoyant algorithm, which is O(log n log m)-competitive, where n is the number of elements and m is the number of sets. This matches the best known result for the classic online set cover (a special case of non-clairvoyant SCD). Moreover, clairvoyance does not allow for significant improvement - we present lower bounds of ?(?{log n}) and ?(?{log m}) for SCD which apply for the clairvoyant case.
In addition, the competitiveness of our algorithm does not depend on the number of requests. Such a guarantee on the size of the universe alone was not previously known even for the clairvoyant case - the only previously-known algorithm (due to Carrasco et al.) is clairvoyant, with competitiveness that grows with the number of requests.
For the special case of vertex cover with delay, we show a simpler, deterministic algorithm which is 3-competitive (and also non-clairvoyant)
The Complexity of Online Graph Games
Online computation is a concept to model uncertainty where not all
information on a problem instance is known in advance. An online algorithm
receives requests which reveal the instance piecewise and has to respond with
irrevocable decisions. Often, an adversary is assumed that constructs the
instance knowing the deterministic behavior of the algorithm. From a game
theoretical point of view, the adversary and the online algorithm are players
in a two-player game. By applying this view on combinatorial graph problems,
especially on problems where the solution is a subset of the vertices, we
analyze their complexity. For this, we introduce a framework based on gadget
reductions from 3-Satisfiability and extend it to an online setting where the
graph is a priori known by a map. This is done by identifying a set of rules
for the reductions and providing schemes for gadgets. The extension of the
framework to the online setting enable reductions from TQBF. We provide example
reductions to the well-known problems Vertex Cover, Independent Set and
Dominating Set and prove that they are PSPACE-complete. Thus, this paper
establishes that the online version with a map of NP-complete graph problems
form a large class of PSPACE-complete problems
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
A Bicriteria Approximation for the Reordering Buffer Problem
In the reordering buffer problem (RBP), a server is asked to process a
sequence of requests lying in a metric space. To process a request the server
must move to the corresponding point in the metric. The requests can be
processed slightly out of order; in particular, the server has a buffer of
capacity k which can store up to k requests as it reads in the sequence. The
goal is to reorder the requests in such a manner that the buffer constraint is
satisfied and the total travel cost of the server is minimized. The RBP arises
in many applications that require scheduling with a limited buffer capacity,
such as scheduling a disk arm in storage systems, switching colors in paint
shops of a car manufacturing plant, and rendering 3D images in computer
graphics.
We study the offline version of RBP and develop bicriteria approximations.
When the underlying metric is a tree, we obtain a solution of cost no more than
9OPT using a buffer of capacity 4k + 1 where OPT is the cost of an optimal
solution with buffer capacity k. Constant factor approximations were known
previously only for the uniform metric (Avigdor-Elgrabli et al., 2012). Via
randomized tree embeddings, this implies an O(log n) approximation to cost and
O(1) approximation to buffer size for general metrics. Previously the best
known algorithm for arbitrary metrics by Englert et al. (2007) provided an
O(log^2 k log n) approximation without violating the buffer constraint.Comment: 13 page
Adding Isolated Vertices Makes some Online Algorithms Optimal
An unexpected difference between online and offline algorithms is observed.
The natural greedy algorithms are shown to be worst case online optimal for
Online Independent Set and Online Vertex Cover on graphs with 'enough' isolated
vertices, Freckle Graphs. For Online Dominating Set, the greedy algorithm is
shown to be worst case online optimal on graphs with at least one isolated
vertex. These algorithms are not online optimal in general. The online
optimality results for these greedy algorithms imply optimality according to
various worst case performance measures, such as the competitive ratio. It is
also shown that, despite this worst case optimality, there are Freckle graphs
where the greedy independent set algorithm is objectively less good than
another algorithm. It is shown that it is NP-hard to determine any of the
following for a given graph: the online independence number, the online vertex
cover number, and the online domination number.Comment: A footnote in the .tex file didn't show up in the last version. This
was fixe
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