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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.

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. Advice complexity is a recently introduced method for a fine-grained analysis of the hardness of online problems. We consider the advice complexity of disjoint path allocation, measured in the length L of the path. We show that asking for a bit of advice for every edge is necessary to be optimal and give online algorithms with advice achieving a constant competitive ratio using much less advice. Furthermore, we consider the case of using less than log log L advice bits, where we prove almost matching lower and upper bounds on the competitive ratio. In the latter case, we moreover show that randomness is as powerful as advice by designing a barely random online algorithm achieving almost the same competitive ratio.

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 − δ), for any δ > 0.

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