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.

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Abstract. In this work, we study the advice complexity of the online minimum Steiner tree problem (ST). Given a (known) graph G = (V, E) endowed with a weight function on the edges, a set of N terminals are revealed in a step-wise manner. The algorithm maintains a sub-graph of chosen edges, and at each stage, chooses more edges from G to its solution such that the terminals revealed so far are connected in it. In the standard online setting this problem was studied and a tight bound of O(log(N )) on its competitive ratio is known. Here, we study the power of non-uniform advice and fully characterize it. As a first result we show that using q · log(|V |) advice bits, where 0 ≤ q ≤ N − 1 it is possible to obtain an algorithm with a competitive ratio of O(log(N/q). We then show a matching lower bound for all values of q, and thus settle the question.

Abstract. We present a new framework for proving fully black-box separations and lower bounds. We prove a general theorem that facilitates the proofs of fully black-box lower bounds from a one-way function (OWF).Loosely speaking, our theorem says that in order to prove that a fully black-box construction does not securely construct a cryptographic primitive Q (e.g., a pseudo-random generator or a universal one-way hash function) from a OWF, it is enough to come up with a large enough set of functions F and a parameterized oracle (i.e., an oracle that is defined for every f ∈ {0, 1} n → {0, 1} n ) such that O f breaks the security of the construction when instantiated with f and the oracle satisfies two local properties.Our main application of the theorem is a lower bound of Ω(n/ log(n)) on the number of calls made by any fully black-box construction of a universal one-way hash function (UOWHF) from a general one-way function. The bound holds even when the OWF is regular, in which case it matches to a recent construction of Barhum and Maurer [4].