23 research outputs found
Online Bin Packing with Advice
We consider the online bin packing problem under the advice complexity model
where the 'online constraint' is relaxed and an algorithm receives partial
information about the future requests. We provide tight upper and lower bounds
for the amount of advice an algorithm needs to achieve an optimal packing. We
also introduce an algorithm that, when provided with log n + o(log n) bits of
advice, achieves a competitive ratio of 3/2 for the general problem. This
algorithm is simple and is expected to find real-world applications. We
introduce another algorithm that receives 2n + o(n) bits of advice and achieves
a competitive ratio of 4/3 + {\epsilon}. Finally, we provide a lower bound
argument that implies that advice of linear size is required for an algorithm
to achieve a competitive ratio better than 9/8.Comment: 19 pages, 1 figure (2 subfigures
Online Bin Packing with Advice
We consider the online bin packing problem under the advice complexity model where the "online constraint" is relaxed and an algorithm receives partial information about the future requests. We provide tight upper and lower bounds for the amount of advice an algorithm needs to achieve an optimal packing. We also introduce an algorithm that, when provided with log(n)+o(log(n)) bits of advice, achieves a competitive ratio of 3/2 for the general problem. This algorithm is simple and is expected to find real-world applications. We introduce another algorithm that receives 2n+o(n) bits of advice and achieves a competitive ratio of 4/3+e. Finally, we provide a lower bound argument that implies that advice of linear size is required for an algorithm to achieve a competitive ratio better than 9/8
On the Power of Advice and Randomization for Online Bipartite Matching
While randomized online algorithms have access to a sequence of uniform
random bits, deterministic online algorithms with advice have access to a
sequence of advice bits, i.e., bits that are set by an all powerful oracle
prior to the processing of the request sequence. Advice bits are at least as
helpful as random bits, but how helpful are they? In this work, we investigate
the power of advice bits and random bits for online maximum bipartite matching
(MBM).
The well-known Karp-Vazirani-Vazirani algorithm is an optimal randomized
-competitive algorithm for \textsc{MBM} that requires access
to uniform random bits. We show that
advice bits are necessary and
sufficient in order to obtain a
-competitive deterministic advice algorithm. Furthermore, for a
large natural class of deterministic advice algorithms, we prove that
advice bits are required in order to improve on the
-competitiveness of the best deterministic online algorithm, while
it is known that bits are sufficient.
Last, we give a randomized online algorithm that uses random bits, for
integers , and a competitive ratio that approaches
very quickly as is increasing. For example if , then the difference
between and the achieved competitive ratio is less than
Lower bounds for several online variants of bin packing
We consider several previously studied online variants of bin packing and
prove new and improved lower bounds on the asymptotic competitive ratios for
them. For that, we use a method of fully adaptive constructions. In particular,
we improve the lower bound for the asymptotic competitive ratio of online
square packing significantly, raising it from roughly 1.68 to above 1.75.Comment: WAOA 201
Online Computation with Untrusted Advice
The advice model of online computation captures a setting in which the
algorithm is given some partial information concerning the request sequence.
This paradigm allows to establish tradeoffs between the amount of this
additional information and the performance of the online algorithm. However, if
the advice is corrupt or, worse, if it comes from a malicious source, the
algorithm may perform poorly. In this work, we study online computation in a
setting in which the advice is provided by an untrusted source. Our objective
is to quantify the impact of untrusted advice so as to design and analyze
online algorithms that are robust and perform well even when the advice is
generated in a malicious, adversarial manner. To this end, we focus on
well-studied online problems such as ski rental, online bidding, bin packing,
and list update. For ski-rental and online bidding, we show how to obtain
algorithms that are Pareto-optimal with respect to the competitive ratios
achieved; this improves upon the framework of Purohit et al. [NeurIPS 2018] in
which Pareto-optimality is not necessarily guaranteed. For bin packing and list
update, we give online algorithms with worst-case tradeoffs in their
competitiveness, depending on whether the advice is trusted or not; this is
motivated by work of Lykouris and Vassilvitskii [ICML 2018] on the paging
problem, but in which the competitiveness depends on the reliability of the
advice. Furthermore, we demonstrate how to prove lower bounds, within this
model, on the tradeoff between the number of advice bits and the
competitiveness of any online algorithm. Last, we study the effect of
randomization: here we show that for ski-rental there is a randomized algorithm
that Pareto-dominates any deterministic algorithm with advice of any size. We
also show that a single random bit is not always inferior to a single advice
bit, as it happens in the standard model
On the List Update Problem with Advice
We study the online list update problem under the advice model of
computation. Under this model, an online algorithm receives partial information
about the unknown parts of the input in the form of some bits of advice
generated by a benevolent offline oracle. We show that advice of linear size is
required and sufficient for a deterministic algorithm to achieve an optimal
solution or even a competitive ratio better than . On the other hand, we
show that surprisingly two bits of advice are sufficient to break the lower
bound of on the competitive ratio of deterministic online algorithms and
achieve a deterministic algorithm with a competitive ratio of . In this
upper-bound argument, the bits of advice determine the algorithm with smaller
cost among three classical online algorithms, TIMESTAMP and two members of the
MTF2 family of algorithms. We also show that MTF2 algorithms are
-competitive
Optimal Online Edge Coloring of Planar Graphs with Advice
Using the framework of advice complexity, we study the amount of knowledge
about the future that an online algorithm needs to color the edges of a graph
optimally, i.e., using as few colors as possible. For graphs of maximum degree
, it follows from Vizing's Theorem that bits of
advice suffice to achieve optimality, where is the number of edges. We show
that for graphs of bounded degeneracy (a class of graphs including e.g. trees
and planar graphs), only bits of advice are needed to compute an optimal
solution online, independently of how large is. On the other hand, we
show that bits of advice are necessary just to achieve a
competitive ratio better than that of the best deterministic online algorithm
without advice. Furthermore, we consider algorithms which use a fixed number of
advice bits per edge (our algorithm for graphs of bounded degeneracy belongs to
this class of algorithms). We show that for bipartite graphs, any such
algorithm must use at least bits of advice to achieve
optimality.Comment: CIAC 201