13 research outputs found
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
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
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
Incorporating capacitative constraint to the preference-based conference scheduling via domain transformation approach
No AbstractKeywords: conference scheduling; domain transformation approach; capacity optimizatio
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
A Randomness Threshold for Online Bipartite Matching, via Lossless Online Rounding
Over three decades ago, Karp, Vazirani and Vazirani (STOC'90) introduced the
online bipartite matching problem. They observed that deterministic algorithms'
competitive ratio for this problem is no greater than , and proved that
randomized algorithms can do better. A natural question thus arises: \emph{how
random is random}? i.e., how much randomness is needed to outperform
deterministic algorithms? The \textsc{ranking} algorithm of Karp et
al.~requires random bits, which, ignoring polylog terms,
remained unimproved. On the other hand, Pena and Borodin (TCS'19) established a
lower bound of random bits for any
competitive ratio.
We close this doubly-exponential gap, proving that, surprisingly, the lower
bound is tight. In fact, we prove a \emph{sharp threshold} of random bits for the randomness necessary and sufficient to
outperform deterministic algorithms for this problem, as well as its
vertex-weighted generalization. This implies the same threshold for the advice
complexity (nondeterminism) of these problems.
Similar to recent breakthroughs in the online matching literature, for
edge-weighted matching (Fahrbach et al.~FOCS'20) and adwords (Huang et
al.~FOCS'20), our algorithms break the barrier of by randomizing matching
choices over two neighbors. Unlike these works, our approach does not rely on
the recently-introduced OCS machinery, nor the more established randomized
primal-dual method. Instead, our work revisits a highly-successful online
design technique, which was nonetheless under-utilized in the area of online
matching, namely (lossless) online rounding of fractional algorithms. While
this technique is known to be hopeless for online matching in general, we show
that it is nonetheless applicable to carefully designed fractional algorithms
with additional (non-convex) constraints
Online algorithms with advice for bin packing and scheduling problems
We consider the setting of online computation with advice and study the bin packing problem and a number of scheduling problems. We show that it is possible, for any of these problems, to arbitrarily approach a competitive ratio of 1 with only a constant number of bits of advice per request. For the bin packing problem, we give an online algorithm with advice that is (1+ε)-competitive and uses O(1εlog 1ε) bits of advice per request. For scheduling on m identical machines, with the objective function of any of makespan, machine covering and the minimization of the ℓp norm, p>1, we give similar results. We give online algorithms with advice which are (1+ε)-competitive ((1/(1-ε))-competitive for machine covering) and also use O(1εlog 1ε) bits of advice per request. We complement our results by giving a lower bound that shows that for any online algorithm with advice to be optimal, for any of the above scheduling problems, a non-constant number (namely, at least (1-2mn)log m, where n is the number of jobs and m is the number of machines) of bits of advice per request is needed