3,372 research outputs found

    The Advice Complexity of a Class of Hard Online Problems

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    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(1+(c1)c1/cc)n=Θ(n/c)\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(logn)O(\log n)) to achieve a competitive ratio of cc. 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 Θ(n/c)\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 Ω(n/c)\Omega (n/c) and an upper bound of O((nlogc)/c)O((n\log c)/c) on the advice complexity. We improve on the upper bound by a factor of logc\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

    Knapsack problems in products of groups

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    The classic knapsack and related problems have natural generalizations to arbitrary (non-commutative) groups, collectively called knapsack-type problems in groups. We study the effect of free and direct products on their time complexity. We show that free products in certain sense preserve time complexity of knapsack-type problems, while direct products may amplify it. Our methods allow to obtain complexity results for rational subset membership problem in amalgamated free products over finite subgroups.Comment: 15 pages, 5 figures. Updated to include more general results, mostly in Section

    Dynamic Complexity Meets Parameterised Algorithms

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    Dynamic Complexity studies the maintainability of queries with logical formulas in a setting where the underlying structure or database changes over time. Most often, these formulas are from first-order logic, giving rise to the dynamic complexity class DynFO. This paper investigates extensions of DynFO in the spirit of parameterised algorithms. In this setting structures come with a parameter k and the extensions allow additional "space" of size f(k) (in the form of an additional structure of this size) or additional time f(k) (in the form of iterations of formulas) or both. The resulting classes are compared with their non-dynamic counterparts and other classes. The main part of the paper explores the applicability of methods for parameterised algorithms to this setting through case studies for various well-known parameterised problems

    Optimal Online Edge Coloring of Planar Graphs with Advice

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    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 Δ\Delta, it follows from Vizing's Theorem that O(mlogΔ)O(m\log \Delta) bits of advice suffice to achieve optimality, where mm 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)O(m) bits of advice are needed to compute an optimal solution online, independently of how large Δ\Delta is. On the other hand, we show that Ω(m)\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 Ω(mlogΔ)\Omega(m\log \Delta) bits of advice to achieve optimality.Comment: CIAC 201
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