816,951 research outputs found
The Online Disjoint Set Cover Problem and its Applications
Given a universe of elements and a collection of subsets
of , the maximum disjoint set cover problem (DSCP) is to
partition into as many set covers as possible, where a set cover
is defined as a collection of subsets whose union is . We consider the
online DSCP, in which the subsets arrive one by one (possibly in an order
chosen by an adversary), and must be irrevocably assigned to some partition on
arrival with the objective of minimizing the competitive ratio. The competitive
ratio of an online DSCP algorithm is defined as the maximum ratio of the
number of disjoint set covers obtained by the optimal offline algorithm to the
number of disjoint set covers obtained by across all inputs. We propose an
online algorithm for solving the DSCP with competitive ratio . We then
show a lower bound of on the competitive ratio for any
online DSCP algorithm. The online disjoint set cover problem has wide ranging
applications in practice, including the online crowd-sourcing problem, the
online coverage lifetime maximization problem in wireless sensor networks, and
in online resource allocation problems.Comment: To appear in IEEE INFOCOM 201
Online Disjoint Set Cover Without Prior Knowledge
The disjoint set cover (DSC) problem is a fundamental combinatorial optimization problem concerned with partitioning the (hyper)edges of a hypergraph into (pairwise disjoint) clusters so that the number of clusters that cover all nodes is maximized. In its online version, the edges arrive one-by-one and should be assigned to clusters in an irrevocable fashion without knowing the future edges. This paper investigates the competitiveness of online DSC algorithms. Specifically, we develop the first (randomized) online DSC algorithm that guarantees a poly-logarithmic (O(log^{2} n)) competitive ratio without prior knowledge of the hypergraph\u27s minimum degree. On the negative side, we prove that the competitive ratio of any randomized online DSC algorithm must be at least Omega((log n)/(log log n)) (even if the online algorithm does know the minimum degree in advance), thus establishing the first lower bound on the competitive ratio of randomized online DSC algorithms
Delaying Decisions and Reservation Costs
We study the Feedback Vertex Set and the Vertex Cover problem in a natural
variant of the classical online model that allows for delayed decisions and
reservations. Both problems can be characterized by an obstruction set of
subgraphs that the online graph needs to avoid. In the case of the Vertex Cover
problem, the obstruction set consists of an edge (i.e., the graph of two
adjacent vertices), while for the Feedback Vertex Set problem, the obstruction
set contains all cycles.
In the delayed-decision model, an algorithm needs to maintain a valid partial
solution after every request, thus allowing it to postpone decisions until the
current partial solution is no longer valid for the current request.
The reservation model grants an online algorithm the new and additional
option to pay a so-called reservation cost for any given element in order to
delay the decision of adding or rejecting it until the end of the instance.
For the Feedback Vertex Set problem, we first analyze the variant with only
delayed decisions, proving a lower bound of and an upper bound of on
the competitive ratio. Then we look at the variant with both delayed decisions
and reservation. We show that given bounds on the competitive ratio of a
problem with delayed decisions impliy lower and upper bounds for the same
problem when adding the option of reservations. This observation allows us to
give a lower bound of and an upper bound of
for the Feedback Vertex Set problem. Finally, we show
that the online Vertex Cover problem, when both delayed decisions and
reservations are allowed, is -competitive, where
is the reservation cost per reserved vertex.Comment: 14 Pages, submitte
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
The Advice Complexity of a Class of Hard Online Problems
The advice complexity of an online problem is a measure of how much knowledge
of the future an online algorithm needs in order to achieve a certain
competitive ratio. Using advice complexity, we define the first online
complexity class, AOC. The class includes independent set, vertex cover,
dominating set, and several others as complete problems. AOC-complete problems
are hard, since a single wrong answer by the online algorithm can have
devastating consequences. For each of these problems, we show that
bits of advice are
necessary and sufficient (up to an additive term of ) to achieve a
competitive ratio of .
The results are obtained by introducing a new string guessing problem related
to those of Emek et al. (TCS 2011) and B\"ockenhauer et al. (TCS 2014). It
turns out that this gives a powerful but easy-to-use method for providing both
upper and lower bounds on the advice complexity of an entire class of online
problems, the AOC-complete problems.
Previous results of Halld\'orsson et al. (TCS 2002) on online independent
set, in a related model, imply that the advice complexity of the problem is
. Our results improve on this by providing an exact formula for
the higher-order term. For online disjoint path allocation, B\"ockenhauer et
al. (ISAAC 2009) gave a lower bound of and an upper bound of
on the advice complexity. We improve on the upper bound by a
factor of . For the remaining problems, no bounds on their advice
complexity were previously known.Comment: Full paper to appear in Theory of Computing Systems. A preliminary
version appeared in STACS 201
Online Class Cover Problem
In this paper, we study the online class cover problem where a (finite or
infinite) family of geometric objects and a set of red
points in are given a prior, and blue points from
arrives one after another. Upon the arrival of a blue point, the online
algorithm must make an irreversible decision to cover it with objects from
that do not cover any points of . The objective of the
problem is to place the minimum number of objects. When consists of
all possible translates of a square in , we prove that the
competitive ratio of any deterministic online algorithm is . On the other hand, when the objects are all possible translates of a
rectangle in , we propose an -competitive
deterministic algorithm for the problem.Comment: 27 pages, 23 figure
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