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 ⌊1.13746⋅log2(n)−0.49887⌋ 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 ⌊log2 n⌋+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 $O(\sqrt{n})$ advice bits are sufficient for coloring any bipartite graph on n vertices with at most ⌈log2 n⌉ 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 − 2 is tight. For the proper coloring, n − 2k bits are necessary and n − 2 bits are sufficient, while for the conflict-free coloring case n − 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 $n$ is the number of requests, we show that: (1) A paging algorithm needs $\Omega(n)$ bits of advice to achieve a competitive ratio better than $H_k=\Omega(\log k)$, where $k$ is the cache size. Previously, it was only known that $\Omega(n)$ bits of advice were necessary to achieve a constant competitive ratio smaller than $5/4$. (2) Every $O(n^{1-\varepsilon})$-competitive vertex coloring algorithm must use $\Omega(n\log n)$ bits of advice. Previously, it was only known that $\Omega(n\log n)$ bits of advice were necessary to be optimal. For certain online problems, including the MTS, $k$-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 $O(\log k)$-competitive $k$-server algorithm with advice complexity $o(n)$ if and only if there exists a randomized $O(\log k)$-competitive $k$-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 $\Delta$, 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

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

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 $\log\left(1+(c-1)^{c-1}/c^{c}\right)n=\Theta (n/c)$ 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

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 $15/14$. On the other hand, we show that surprisingly two bits of advice are sufficient to break the lower bound of $2$ on the competitive ratio of deterministic online algorithms and achieve a deterministic algorithm with a competitive ratio of $5/3$. 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 $2.5$-competitive