Sacha Krug

The string guessing problem as a method to prove lower bounds on the advice complexity

The advice complexity of an online problem describes the additional information both necessary and sufficient for online algorithms to compute solutions of a certain quality. In this model, an oracle inspects the input before it is processed by an online algorithm. Depending on the input string, the oracle prepares an advice bit string that is accessed sequentially by the algorithm. The number of advice bits that are read to achieve some specific solution quality can then serve as a fine-grained complexity measure. The main contribution of this paper is to study a powerful method for proving lower bounds on the number of advice bits necessary. To this end, we consider the string guessing problem as a generic online problem and show a lower bound on the number of advice bits needed to obtain a good solution. We use special reductions from string guessing to improve the best known lower bound for the online set cover problem and to give a lower bound on the advice complexity of the online maximum clique problem.

The String Guessing Problem as a Method to Prove Lower Bounds on the Advice Complexity

The advice complexity of an online problem describes the additional information both necessary and sufficient for online algorithms to compute solutions of a certain quality. In this model, an oracle inspects the input before it is processed by an online algorithm. Depending on the input string, the oracle prepares an advice bit string that is accessed sequentially by the algorithm. The number of advice bits that are read to achieve some specific solution quality can then serve as a fine-grained complexity measure. The main contribution of this paper is to study a powerful method for proving lower bounds on the number of advice bits necessary. To this end, we consider the string guessing problem as a generic online problem and show a lower bound on the number of advice bits needed to obtain a good solution. We use special reductions from string guessing to improve the best known lower bound for the online set cover problem and to give a lower bound on the advice complexity of the online maximum clique problem.

On the Power of Advice and Randomization for the Disjoint Path Allocation Problem

In the disjoint path allocation problem, we consider a path of L + 1 vertices, representing the nodes in a communication network. Requests for an unbounded-time communication between pairs of vertices arrive in an online fashion and a central authority has to decide which of these calls to admit. The constraint is that each edge in the path can serve only one call and the goal is to admit as many calls as possible.

On the advice complexity of the online L ( 2 , 1 ) -coloring problem on paths and cycles

In an L ( 2 , 1 ) -coloring of a graph, the vertices are colored with colors from an ordered set such that neighboring vertices get colors that have distance at least 2 and vertices at distance 2 in the graph get different colors. We consider the problem of finding an L ( 2 , 1 ) -coloring using a minimum range of colors in an online setting where the vertices arrive in consecutive time steps together with information about their neighbors and vertices at distance 2 among the previously revealed vertices. For this, we restrict our attention to paths and cycles.Offline, paths can easily be colored within the range { 0 , ? , 4 } of colors. We prove that, considering deterministic algorithms in an online setting, the range { 0 , ? , 6 } is necessary and sufficient while a simple greedy strategy needs range { 0 , ? , 7 } .Advice complexity is a recently developed framework to measure the complexity of online problems. The idea is to measure how many bits of advice about the yet unknown parts of the input an online algorithm needs to compute a solution of a certain quality. We show a sharp threshold on the advice complexity of the online L ( 2 , 1 ) -coloring problem on paths and cycles. While achieving color range { 0 , ? , 6 } does not need any advice, improving over this requires a number of advice bits that is linear in the size of the input. Thus, the L ( 2 , 1 ) -coloring problem is the first known example of an online problem for which sublinear advice does not help.We further use our advice complexity results to prove that no randomized online algorithm can achieve a better expected competitive ratio than 5 4 ( 1 - ? ) , for any ? 0 .

On the Advice Complexity of the Online L(2,1)-Coloring Problem on Paths and Cycles

In an L(2,1)-coloring of a graph, the vertices are colored with colors from an ordered set such that neighboring vertices get colors that have distance at least 2 and vertices at distance 2 in the graph get different colors. We consider the problem of finding an L(2,1)-coloring using a minimum range of colors in an online setting where the vertices arrive in consecutive time steps together with information about their neighbors and vertices at distance two among the previously revealed vertices. For this, we restrict our attention to paths and cycles.

Improved analysis of the online set cover problem with advice

Abstract We study the advice complexity of an online version of the set cover problem. The goal is to quantify the information that online algorithms for this problem need to be supplied with to compute high-quality solutions and to overcome the drawback of not knowing future requests. This concept was successfully applied to many prominent online problems in the past while trying to capture the essence of “what makes an online problem hard.” The online set cover problem was introduced by Alon et al. (2009) [2] : for a ground set of size n and a set family of m subsets of the ground set, we obtain bounds in both n and m . We show that a linear number (with respect to both n and m ) of advice bits is both sufficient and necessary to perform optimally. Furthermore, we prove that O ( ( n log ⁡ c ) / c ) bits are sufficient to design a c -competitive online algorithm, and this bound is tight up to a factor of O ( log ⁡ c ) . We further give upper and lower bounds for achieving c -competitiveness with respect to m . Finally, we analyze the advice complexity of the problem with respect to some natural parameters, i.e., measurable properties of the inputs.

On energy-efficient computations with advice

Online problems have always played an important role in computer science. Here, not the whole input is known at the beginning, but it is only revealed gradually. These problems frequently occur in practice, and therefore the performance of algorithms for such problems is of great theoretical and practical interest. One such online problem is energy management in electronic devices, e. g., smartphones. As such a device is usually not being used permanently, it is reasonable to change to a lower-energy state (like hibernation) after a certain idle time. Resuming from hibernation, however, also needs a certain amount of energy; therefore, hibernation should only happen if the idle period is long.

Towards using the history in online computation with advice

Recently, advice complexity has been introduced as a new framework to analyze online algorithms. There, an online algorithm has access to an infinite binary advice tape during the computation. The contents of this tape were prepared beforehand by an omniscient oracle. One is interested in analyzing the number of accessed advice bits necessary and/or sufficient to achieve a certain solution quality. Among others, the bit guessing problem was analyzed in this framework. Here, an algorithm needs to guess a binary string bit by bit, either with or without getting immediate feedback after each bit. The bit guessing problem can be used to obtain lower bounds on the advice complexity of a variety of other online problems. In this paper, we analyze the difference between the two bit guessing variants. More precisely, we show that getting feedback after each request helps save advice bits when we allow errors to be made. This is by no means obvious – for optimality, both problem versions need the same amount of advice, and without advice, knowing the history does not help at all.

On the approximation ratio of the path matching christofides algorithm

The traveling salesman problem (TSP) is one of the most fundamental optimization problems. We consider the β -metric traveling salesman problem (Δβ -TSP), i.e., the TSP restricted to graphs satisfying the β -triangle inequality c ({v ,w })≤β (c ({v ,u })+c (u ,w })), for some cost function c and any three vertices u ,v ,w . The well-known path matching Christofides algorithm (PMCA) guarantees an approximation ratio of

The String Guessing Problem as a Method to Prove Lower Bounds on the Advice Complexity.

\frac{3}{2}\beta^2