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 $\log\left(1+(c-1)^{c-1}/c^{c}\right)n=\Theta (n/c)$ bits of advice are necessary and sufficient (up to an additive term of $O(\log n)$) to achieve a competitive ratio of $c$. 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 $\Theta (n/c)$. 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 $\Omega (n/c)$ and an upper bound of $O((n\log c)/c)$ on the advice complexity. We improve on the upper bound by a factor of $\log c$. 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

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 $\Delta$, it follows from Vizing's Theorem that $O(m\log \Delta)$ bits of advice suffice to achieve optimality, where $m$ 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 $O(m)$ bits of advice are needed to compute an optimal solution online, independently of how large $\Delta$ is. On the other hand, we show that $\Omega (m)$ 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 $\Omega(m\log \Delta)$ 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

Advice Complexity of the Online Induced Subgraph Problem

Several well-studied graph problems aim to select a largest (or smallest) induced subgraph with a given property of the input graph. Examples of such problems include maximum independent set, maximum planar graph, and many others. We consider these problems, where the vertices are presented online. With each vertex, the online algorithm must decide whether to include it into the constructed subgraph, based only on the subgraph induced by the vertices presented so far. We study the properties that are common to all these problems by investigating the generalized problem: for a hereditary property \pty, find some maximal induced subgraph having \pty. We study this problem from the point of view of advice complexity. Using a result from Boyar et al. [STACS 2015], we give a tight trade-off relationship stating that for inputs of length n roughly n/c bits of advice are both needed and sufficient to obtain a solution with competitive ratio c, regardless of the choice of \pty, for any c (possibly a function of n). Surprisingly, a similar result cannot be obtained for the symmetric problem: for a given cohereditary property \pty, find a minimum subgraph having \pty. We show that the advice complexity of this problem varies significantly with the choice of \pty. We also consider preemptive online model, where the decision of the algorithm is not completely irreversible. In particular, the algorithm may discard some vertices previously assigned to the constructed set, but discarded vertices cannot be reinserted into the set again. We show that, for the maximum induced subgraph problem, preemption cannot help much, giving a lower bound of $\Omega(n/(c^2\log c))$ bits of advice needed to obtain competitive ratio $c$, where $c$ is any increasing function bounded by \sqrt{n/log n}. We also give a linear lower bound for c close to 1

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

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 $(1-\frac{1}{e})$-competitive algorithm for \textsc{MBM} that requires access to $\Theta(n \log n)$ uniform random bits. We show that $\Omega(\log(\frac{1}{\epsilon}) n)$ advice bits are necessary and $O(\frac{1}{\epsilon^5} n)$ sufficient in order to obtain a $(1-\epsilon)$-competitive deterministic advice algorithm. Furthermore, for a large natural class of deterministic advice algorithms, we prove that $\Omega(\log \log \log n)$ advice bits are required in order to improve on the $\frac{1}{2}$-competitiveness of the best deterministic online algorithm, while it is known that $O(\log n)$ bits are sufficient. Last, we give a randomized online algorithm that uses $c n$ random bits, for integers $c \ge 1$, and a competitive ratio that approaches $1-\frac{1}{e}$ very quickly as $c$ is increasing. For example if $c = 10$, then the difference between $1-\frac{1}{e}$ and the achieved competitive ratio is less than $0.0002$

Online Computation with Untrusted Advice

The advice model of online computation captures the setting in which the online 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, unlike real life in which advice is a recommendation that we can choose to follow or to ignore based on trustworthiness, in the current advice model, the online algorithm treats it as infallible. This means that 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

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, núm. 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 “minimalistic online algorithms”.Peer ReviewedPostprint (author's final draft