Lene M. Favrholdt

Antonios Antoniadis, Joan Boyar, Marek Eliáš et al.

Paging is a prototypical problem in the area of online algorithms. It has also played a central role in the development of learning-augmented algorithms -- a recent line of research that aims to ameliorate the shortcomings of classical worst-case analysis by giving algorithms access to predictions. Such predictions can typically be generated using a machine learning approach, but they are inherently imperfect. Previous work on learning-augmented paging has investigated predictions on (i) when the current page will be requested again (reoccurrence predictions), (ii) the current state of the cache in an optimal algorithm (state predictions), (iii) all requests until the current page gets requested again, and (iv) the relative order in which pages are requested. We study learning-augmented paging from the new perspective of requiring the least possible amount of predicted information. More specifically, the predictions obtained alongside each page request are limited to one bit only. We consider two natural such setups: (i) discard predictions, in which the predicted bit denotes whether or not it is ``safe'' to evict this page, and (ii) phase predictions, where the bit denotes whether the current page will be requested in the next phase (for an appropriate partitioning of the input into phases). We develop algorithms for each of the two setups that satisfy all three desirable properties of learning-augmented algorithms -- that is, they are consistent, robust and smooth -- despite being limited to a one-bit prediction per request. We also present lower bounds establishing that our algorithms are essentially best possible.

Magnus Berg, Joan Boyar, Lene M. Favrholdt et al.

With the developments in machine learning, there has been a surge in interest and results focused on algorithms utilizing predictions, not least in online algorithms where most new results incorporate the prediction aspect for concrete online problems. While the structural computational hardness of problems with regards to time and space is quite well developed, not much is known about online problems where time and space resources are typically not in focus. Some information-theoretical insights were gained when researchers considered online algorithms with oracle advice, but predictions of uncertain quality is a very different matter. We initiate the development of a complexity theory for online problems with predictions, considering minimization problems and one prediction bit per request. Based on the most generic hard online problem type, string guessing, we define a family of hierarchies of complexity classes (indexed by pairs of error measures) and develop notions of reductions, class membership, hardness, and completeness. Our framework contains all the tools one expects to find when working with complexity, and we illustrate our tools by analyzing problems with different characteristics. In addition, we show that known lower bounds for paging with discard predictions apply directly to all hard problems for each class in the hierarchy based on the canonical pair of error measures. This paging problem is not complete for these classes. Our work also implies corresponding complexity classes for classic online problems without predictions, with the corresponding complete problems.

Joan Boyar, Lene M. Favrholdt, Kim S. Larsen et al.

We consider the problem of forwarding packets arriving online with their destinations in a line network. In each time step, each router can forward one packet along the edge to its right. Each packet that is forwarded arrives at the next router one time step later. Packets are forwarded until they reach their destination. The flow time of a packet is the difference between its release time and the time of its arrival at its destination. The goal is to minimize the maximum flow time. This problem was introduced by Antoniadis et al.~in 2014. They propose a collection of natural algorithms and prove for one, and claim for others, that none of them are $O(1)$-competitive. It was posed as an open problem whether such an algorithm exists. We make the first progress on answering this question. We consider the special case where each packet needs to be forwarded by exactly one or two routers. We show that a greedy algorithm, which was not previously considered for this problem, achieves a competitive ratio of exactly $2-2^{1-k}$, where $k$ is the number of active routers in the network. We also give a general lower bound of $4/3$, even for randomized algorithms.