8,488 research outputs found
Online Coloring of Bipartite Graphs with and without Advice
In the online version of the well-known graph coloring problem, the vertices appear one after the other together with the edges to the already known vertices and have to be irrevocably colored immediately after their appearance. We consider this problem on bipartite, i.e., two-colorable graphs. We prove that at least â1.13746â
log2(n)â0.49887â colors are necessary for any deterministic online algorithm to be able to color any given bipartite graph on n vertices, thus improving on the previously known lower bound of âlog2 nâ+1 for sufficiently large n. Recently, the advice complexity was introduced as a method for a fine-grained analysis of the hardness of online problems. We apply this method to the online coloring problem and prove (almost) tight linear upper and lower bounds on the advice complexity of coloring a bipartite graph online optimally or using 3 colors. Moreover, we prove that advice bits are sufficient for coloring any bipartite graph on n vertices with at most âlog2 nâ colors
On the advice complexity of coloring bipartite graphs and two-colorable hypergraphs
In the online coloring problem the vertices are revealed one by one to an online algorithm, which has to color them immediately as they appear. The advice complexity attempts to measure how much knowledge of the future is neccessary to achieve a given competitive ratio. Here, we examine coloring of bipartite graphs, proper and the conflict-free coloring of k-uniform hypergraphs and we provide lower and upper bounds for the number of bits of advice to achieve the optimal cost. For bipartite graphs the upper bound n â 2 is tight. For the proper coloring, n â 2k bits are necessary and n â 2 bits are sufficient, while for the conflict-free coloring case n â 2 bits of advice are neccessary and sufficient to color optimally if k > 3
Randomization can be as helpful as a glimpse of the future in online computation
We provide simple but surprisingly useful direct product theorems for proving
lower bounds on online algorithms with a limited amount of advice about the
future. As a consequence, we are able to translate decades of research on
randomized online algorithms to the advice complexity model. Doing so improves
significantly on the previous best advice complexity lower bounds for many
online problems, or provides the first known lower bounds. For example, if
is the number of requests, we show that:
(1) A paging algorithm needs bits of advice to achieve a
competitive ratio better than , where is the cache
size. Previously, it was only known that bits of advice were
necessary to achieve a constant competitive ratio smaller than .
(2) Every -competitive vertex coloring algorithm must
use bits of advice. Previously, it was only known that
bits of advice were necessary to be optimal.
For certain online problems, including the MTS, -server, paging, list
update, and dynamic binary search tree problem, our results imply that
randomization and sublinear advice are equally powerful (if the underlying
metric space or node set is finite). This means that several long-standing open
questions regarding randomized online algorithms can be equivalently stated as
questions regarding online algorithms with sublinear advice. For example, we
show that there exists a deterministic -competitive -server
algorithm with advice complexity if and only if there exists a
randomized -competitive -server algorithm without advice.
Technically, our main direct product theorem is obtained by extending an
information theoretical lower bound technique due to Emek, Fraigniaud, Korman,
and Ros\'en [ICALP'09]
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
Online Multi-Coloring with Advice
We consider the problem of online graph multi-coloring with advice.
Multi-coloring is often used to model frequency allocation in cellular
networks. We give several nearly tight upper and lower bounds for the most
standard topologies of cellular networks, paths and hexagonal graphs. For the
path, negative results trivially carry over to bipartite graphs, and our
positive results are also valid for bipartite graphs. The advice given
represents information that is likely to be available, studying for instance
the data from earlier similar periods of time.Comment: IMADA-preprint-c
Online graph coloring against a randomized adversary
Electronic version of an article published as
Online graph coloring against a randomized adversary. "International journal of foundations of computer science", 1 Juny 2018, vol. 29, nĂșm. 4, p. 551-569. DOI:10.1142/S0129054118410058 © 2018 copyright World Scientific Publishing Company. https://www.worldscientific.com/doi/abs/10.1142/S0129054118410058We consider an online model where an adversary constructs a set of 2s instances S instead of one single instance. The algorithm knows S and the adversary will choose one instance from S at random to present to the algorithm. We further focus on adversaries that construct sets of k-chromatic instances. In this setting, we provide upper and lower bounds on the competitive ratio for the online graph coloring problem as a function of the parameters in this model. Both bounds are linear in s and matching upper and lower bound are given for a specific set of algorithms that we call âminimalistic online algorithmsâ.Peer ReviewedPostprint (author's final draft
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
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
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
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