Search

We study the nascent setting of online computation with imperfect advice, in which the online algorithm is enhanced by some prediction encoded in the form of a possibly erroneous binary string. The algorithm is oblivious to the advice error, but defines a desired tolerance, namely an upper bound on the number of erroneous advice bits it can tolerate. This is a model that generalizes the untrusted advice model [Angelopoulos et al. ITCS 2020], in which the performance of the algorithm is only evaluated at the extreme values of error (namely, if the advice has either no errors, or if it is generated adversarially). In this work, we establish connections between games with a lying responder, also known as Rényi-Ulam games, and the design and analysis of online algorithms with imperfect advice. Specifically, we demonstrate how to obtain upper and lower bounds on the competitive ratio for well-studied online problems such as time-series search, online bidding, and fractional knapsack. Our techniques provide the first lower bounds for online problems in this model. We also highlight and exploit connections between competitive analysis with imperfect advice and fault-tolerance in multiprocessor systems. Last, we show how to waive the dependence on the tolerance parameter, by means of resource augmentation and robustification.

Several well-studied online resource allocation problems can be formulated in terms of infinite, increasing sequences of positive values, in which each element is associated with a corresponding allocation value. Examples include problems such as online bidding, searching for a hidden target on an unbounded line, and designing interruptible algorithms based on repeated executions. The performance of the online algorithm, in each of these problems, is measured by the competitive ratio, which describes the multiplicative performance loss due to the absence of full information on the instance. We study such competitive sequencing problems in a setting in which the online algorithm has some (potentially) erroneous information, expressed as a $k$-bit advice string, for some given $k$. We first consider the untrusted advice setting of [Angelopoulos et al, ITCS 2020], in which the objective is to quantify performance considering two extremes: either the advice is either error-free, or it is generated by a (malicious) adversary. Here, we show a Pareto-optimal solution, using a new approach for applying the functional-based lower-bound technique due to [Gal, Israel J. Math. 1972]. Next, we study a nascent noisy advice setting, in which a number of the advice bits may be erroneous; the exact error is unknown to the online algorithm, which only has access to a pessimistic estimate (i.e., an upper bound on this error). We give improved upper bounds, but also the first lower bound on the competitive ratio of an online problem in this setting. To this end, we combine ideas from robust query-based search in arrays, and fault-tolerant contract scheduling. Last, we demonstrate how to apply the above techniques in robust optimization without predictions, and discuss how they can be applicable in the context of more general online problems.

The linear search problem, informally known as the cow path problem, is one of the fundamental problems in search theory. In this problem, an immobile target is hidden at some unknown position on an unbounded line, and a mobile searcher, initially positioned at some specific point of the line called the root, must traverse the line so as to locate the target. The objective is to minimize the worst-case ratio of the distance traversed by the searcher to the distance of the target from the root, which is known as the competitive ratio of the search. In this work we study this problem in a setting in which the searcher has a hint concerning the target. We consider three settings in regards to the nature of the hint: i) the hint suggests the exact position of the target on the line; ii) the hint suggests the direction of the optimal search (i.e., to the left or the right of the root); and iii) the hint is a general k-bit string that encodes some information concerning the target. Our objective is to study the Pareto-efficiency of strategies in this model. Namely, we seek optimal, or near-optimal tradeoffs between the searcher’s performance if the hint is correct (i.e., provided by a trusted source) and if the hint is incorrect (i.e., provided by an adversary)., LIPIcs, Vol. 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021), pages 51:1-51:16

International audience; The advice model of online computation captures the setting in which the online algorithm is given some partial information concerning the request sequence. This paradigm allows to establish tradeoffs between the amount of this additional information and the performance of the online algorithm. However, unlike real life in which advice is a recommendation that we can choose to follow or to ignore based on trustworthiness, in the current advice model, the online algorithm treats it as infallible. This means that if the advice is corrupt or, worse, if it comes from a malicious source, the algorithm may perform poorly. In this work, we study online computation in a setting in which the advice is provided by an untrusted source. Our objective is to quantify the impact of untrusted advice so as to design and analyze online algorithms that are robust and perform well even when the advice is generated in a malicious, adversarial manner. To this end, we focus on well- studied online problems such as ski rental, online bidding, bin packing, and list update. For ski-rental and online bidding, we show how to obtain algorithms that are Pareto-optimal with respect to the competitive ratios achieved; this improves upon the framework of Purohit et al. [NeurIPS 2018] in which Pareto-optimality is not necessarily guaranteed. For bin packing and list update, we give online algorithms with worst-case tradeoffs in their competitiveness, depending on whether the advice is trusted or not; this is motivated by work of Lykouris and Vassilvitskii [ICML 2018] on the paging problem, but in which the competitiveness depends on the reliability of the advice. Furthermore, we demonstrate how to prove lower bounds, within this model, on the tradeoff between the number of advice bits and the competitiveness of any online algorithm. Last, we study the effect of randomization: here we show that for ski-rental there is a randomized algorithm that Pareto-dominates any deterministic algorithm with advice of any size. We also show that a single random bit is not always inferior to a single advice bit, as it happens in the standard model.

International audience; In search problems, a mobile searcher seeks to locate a target that hides in some unknown position of the environment. Such problems are typically considered to be of an on-line nature, in that the input is unknown to the searcher, and the performance of a search strategy is usually analyzed by means of the standard framework of the competitive ratio, which compares the cost incurred by the searcher to an optimal strategy that knows the location of the target. However, one can argue that even for simple search problems, competitive analysis fails to distinguish between strategies which, intuitively, should have different performance in practice. Motivated by the above, in this work we introduce and study measures supplementary to competitive analysis in the context of search problems. In particular, we focus on the well-known problem of linear search, informally known as the cow-path problem, for which there is an infinite number of strategies that achieve an optimal competitive ratio equal to 9. We propose a measure that reflects the rate at which the line is being explored by the searcher, and which can be seen as an extension of the bijective ratio over an uncountable set of requests. Using this measure we show that a natural strategy that explores the line aggressively is optimal among all 9-competitive strategies. This provides, in particular, a strict separation from the competitively optimal doubling strategy, which is much more conservative in terms of exploration. We also provide evidence that this aggressiveness is requisite for optimality, by showing that any optimal strategy must mimic the aggressive strategy in its first few explorations.

We introduce and study the general setting of weighted search in which a number of targets, each with a certain weight, are hidden in a star-like environment that consists of $m$ infinite, concurrent rays, with a common origin. A mobile searcher, initially located at the origin, explores this environment so as to locate a set of targets whose aggregate weight is at least a given value $W$. The cost of the search strategy is defined as the total distance traversed by the searcher, and its performance is measured by the worst-case ratio of the cost incurred by the searcher over the cost of an on optimal, offline strategy with complete access to the instance. This is the first study of a setting that generalizes several problems in search theory: the problem in which only a single target is sought, as well as the problem in which all targets have unit weights. We present and analyze a search strategy of near-optimal performance for the problem at hand. We observe that the classical approaches that rely on geometrically increasing search depths perform rather poorly in the context of weighted search. We bypass this problem by using a strategy that modifies the search depths adaptively, depending on the number of targets located up to the current point in time.

Bin packing is a classic optimization problem with a wide range of applications from load balancing in networks to supply chain management. In this work we study the online variant of the problem, in which a sequence of items of various sizes must be placed into a minimum number of bins of uniform capacity. The online algorithm is enhanced with a (potentially erroneous) prediction concerning the frequency of item sizes in the sequence. We design and analyze online algorithms with efficient tradeoffs between consistency (i.e., the competitive ratio assuming no prediction error) and robustness (i.e., the competitive ratio under adversarial error), and whose performance degrades gently as a function of the prediction error. This is the first theoretical study of online bin packing in the realistic setting of erroneous predictions, as well as the first experimental study in the setting in which the input is generated according to both static and evolving distributions. Previous work on this problem has only addressed the extreme cases with respect to the prediction error, has relied on overly powerful and error-free prediction oracles, and has focused on experimental evaluation based on static input distributions., 28 pages, 4 figures

International audience; Contract scheduling is a general technique that allows to design a system with interruptible capabilities, given an algorithm that is not necessarily interruptible. Previous work on this topic has largely assumed that the interruption is a worst-case deadline that is unknown to the scheduler. In this work, we study the setting in which there is a potentially erroneous prediction concerning the interruption. Specifically, we consider the setting in which the prediction describes the time that the interruption occurs, as well as the setting in which the prediction is obtained as a response to a single or multiple binary queries. For both settings, we investigate tradeoffs between the robustness (i.e., the worst-case performance assuming adversarial prediction) and the consistency (i.e, the performance assuming that the prediction is error-free), both from the side of positive and negative results.